VIA REGIA

Ad

GEOMETRIAM.

THE WAY

TO

GEOMETRY.

Being necessary and usefull,

For

Astronomers. Engineres. Geographers. Architecks. Land-meaters. Carpenters. Sea-men. Paynters. Carvers, &c.

Written in Latine by Peter Ramvs, and now Translated and much enlarged by the Learned M^r. William Bedwell.

LONDON,
Printed by Thomas Cotes, And are to be sold by
Michael Sparke; at the blew Bible in
Greene Arbour, 1636.

TO THE

WORSHIPFVL

M. Iohn Greaves, Professor of

Geometry in Gresham Colledge London;

All happinesse.

SIR,

Your acquaintance with the Author before his death was not long, which I have oft heard you say, you counted your great unhappinesse, but within a short time after, you knew not well whether to count your selfe more happie in that you once knew him, or unhappy in that upon your acquaintance you so suddenly lost him. This his worke then being to come forth to the censorious eye of the world, and as the manner usually is to have some Patronage, I have thought good to dedicate it to your selfe; and that for these two reasons especially.

First, in respect of the sympathy betwixt it, and your studies; Laboures of this nature being usually offered to such persons whose profession is that way setled.

Secondly, for the great love and respect you alwayes shewed to the Author, being indeed a man that would deserve no lesse, humble, void of pride, ever ready to impart his knowledge to others in what kind soever, loving and affecting those that affected learning.

For these respects then, I offer to you this Worke of your so much honoured friend. I my selfe also (as it is no lesse my duty) for his sake striving to make you hereby some part of a requitall, least I should be found guilty of ingratitude, which is a solecisme in manners, if having so fit an opportunity, I should not expresse to the world some Testimonie of love to you, who so much loved him. I desire then (good Sir) your kind acceptance of it, you knowing so well the ability of the Author, and being also able to judge of a Worke of this nature, and in that respect the better able to defend it from the furie of envious Detractours, of which there are not few. Thus with my best wishes to you, as to my much respected friend, I rest.

Yours to be commanded in

any thing that he is able.

Iohn Clerke.

To the Reader.

Friendly Reader, that which is here set forth to thy view, is a Translation out of Ramus. Formerly indeed Translated by one M^r. Thomas Hood, but never before set forth with the Demonstrations and Diagrammes, which being cut before the Authors death, and the Worke it selfe finished, the Coppie I having in mine hands, never had thought for the promulgation of it, but that it should have died with its Author, considering no small prejudice usually attends the printing of dead mens Workes, and wee see the times, the world is now all eare and tongue, the most given with the Athenians, to little else than to heare and tell newes: And if Apelles that skilfull Artist alwayes found somewhat to be amended in those Pictures which he had most curiously drawne; surely much in this Worke might have beene amended if the Authour had lived to refine it, but in that it was onely the first draught, and that he was prevented by death of a second view, though perused by others before the Presse; I was ever unwilling to the publication, but that I was often and much solicited with iteration of strong importunity, and so in the end over-ruled: perswading me from time to time unto it, and that it being finished by the Authour, it was farre better to be published, though with some errours and escapes, than to be onely moths-meat, and so utterly lost. I would have thee, Courteous Reader know, that it is no conceit of the worth of the thing that I should expose the name and credit of the Authour to a publike censure; yet I durst be bold to say, had he lived to have fitted it, and corrected the Presse, the worke would have pointed out the workeman. For I may say, without vaine ostentation, he was a man of worth and note, and there was not that kinde of learning in which he had not some knowledge, but especially for the Easterne tongues, those deepe and profound Studies, in the judgement of the learned, which knew him well, he hath not left his fellow behind him; as his Workes also in Manuscript now extant in the publike Library of the famous Vniversity of Cambridge; do testifie no lesse; for him then being so grave and learned a Divine to meddle with a worke of this nature, he gives thee a reason in his owne following Preface for his principall end and intent of taking this Worke in hand, was not for the deepe and Iudiciall, but for the shallowest skull, the good and profit of the simpler sort, who as it was in the Latine, were able to get little or no benifite from it. Therefore considering the worth of the Authour, and his intent in the Worke. Reade it favourably, and if the faults be not too great, cover them with the mantle of love, and judge charitably offences unwillingly committed, and doe according to the termes of equitie, as thou wouldest be done unto, but it is a common saying, as Printers get Copies for their profit, so Readers often buy and reade for their pleasure; and there is no worke so exactly done that can escape the malevolous disposition of some detracting spirits, to whom I say, as one well, Facilius est unicuivis nostrum aliena curiosè observare: quam proproia negotia rectè agere. It is a great deale more easie to carpe at other mens doings, than to give better of his owne. And as Arist. τό πάσιν ἀρέσαι δυσχερέστατόν ἐστι; omnibus placere difficilimum est. But wherefore, Gentle Reader, should I make any doubt of thy curtesie, and favourable acceptance; for surely there can be nothing more contrary to equitie, than to speake evill of those that have taken paines to doe good, a Pagan would hardly doe this, much lesse I hope any good Christian. Read then, and if by reading, thou reapest any profit, I have my desire, if not, the fault shall be thine owne, reading haply more to judge and censure, than for any good and benefit which otherwise may be received from it; let but the same mind towards thine owne good possesse thee in reading it, as did the Author in writing it, and there shall be no neede to doubt of thy profit by it.

Thine in the common

bond of love,

Iohn Clerke.

The Authors Preface.

Two things, I feare me, will here be objected against me: The one concerneth my selfe, directly: The other mine Author, and the worke I have taken in hand the translating of him. Concerning my selfe, I suppose, some will aske, Why I being a Divine; should meddle or busie my selfe with these prophane studies? Geometry may no way further Divinity, and therefore is no fit study for a Divine? This objection seemeth to smell of Brownisme, that is, of a ranke peevish humour overflowing the stomach of some, whereby they are caused to loath all manner of solid learning, yea of true Divinity it selfe, and therefore it doth not deserve an answer: And this we in our Title before signified. For we have not taken this paines for Turkes and others, who by the lawes of their profession are bound to abandon all manner of learning. But if any man shall propose it, as a question, with a desire of satisfaction, we are ready to answer him to the best of our abilitie. First, that Theologia vera est ars artium & scientia scientiarum, Divinity is the Art of Arts, and Science of Sciences; or Divinity is the Mistresse upon which all Arts and Sciences are to attend as servants and handmaides. And why then not Geometry? But in what place she should follow her, I dare not say: For I am no herald, and therefore I meddle not with precedencie: But if I were, she should be none of the hindermost of her traine.

The Oratour saith, and very truly doubtlesse, That, Omnes artes, quæ ad humanitatē pertinent, habent commune quoddam vinculum, & cognatione quadam inter se continentur. All Arts which pertaine unto humanity, they have a certaine common bond, and are knit together by a kinde of affinity. If then any Arts and Sciences may be thought necessary attendants upon this great Lady; Then surely Geometry amongst the rest must needes be one: For otherwise her traine will be but loose and shattered.

Plato saith τὸν θεὸν ἀκεὶ γεωμετρεῖν, That God doth alwayes worke by Geometry, that is, as the wiseman doth interprete it, Sap. XI. 21. Omnia in mensura & numero & pondere disponere. Dispose all things by measure, and number, and weight: Or, as the learned Plutarch speaketh; He adorneth and layeth out all the parts of the world according to rate, proportion, and similitude. Now who, I pray you, understandeth what these termes meane, but he which hath some meane skill in Geometry? Therefore none but such an one, may be able to declare and teach these things unto others.

How many things are there in holy Scripture which may not well be understood without some meane skill in Geometry? The Fabricke and bignesse of Noah's Arke: The Sciagraphy of the Temple set out by Ezechiel, Who may understand, but he that is skilfull in these Arts? I speake not of many and sundry words both in the New and Old Testaments, whose genuine and proper signification is merely Geometricall: And cannot well be conceived but of a Geometer.

And here, that I may speake it without offence, I would have it observed, how many men, much magnified for learning, not onely in their speeches, which alwayes are not premeditated, but even in their writings, exposed to the view and censure of all men, doe often paralogizein, speake much, and little to the purpose. This they could not so easily and often doe, if they had beene but meanely practised in these kinde of studies. Wherefore that Epigramme which was used to be written over their Philosophy Schoole doores, οὐδῆις ἀγεωμέτρητος εἴσιτω, No man ignorant of Geometry come within these doores: Now written over our Divinitie Schooles. And if any man shall thinke this an hard sentence, let him heare what Saint Augustine saith in the same case, Nemo ad divinarum humanarumq; rerum cognitionem accedat, nisi prius annumerandi artem addiscat: Let no man come neither within the Divinity nor Philosophy Schooles, except he have first learned Arithmeticke. Now that the one of these Arts cannot be learned without the other; Euclide our great Master, who made but one of both, hath sufficiently demonstrated.

If I should alledge the like practise of famous Divines, greatly admired for their great skill in this profession, as T. Peckham Arch-Bishop of Canterbury, Maurolycus Bishop of Messana in Sicilia, Cusanus Cardinall of Rome, and many others, before indifferent judges, I am sure I should not be condemned. Who doth not greatly magnifie the grave Seb. Munster, the nimble Ph. Melanchthon, and the noble Bernardino Baldo Abbot of Guastill, and the painefull Barth. Pitiscus of Grunberg, for their knowledge and paines in these Arts and Sciences? And thus much shall at this time suffice, to have spoken unto the first Question: If any shall require further satisfaction, those I referre unto the forenamed Authors, whose authority peradventure may more prevaile with them, then my reasons may.

The next is concerning mine Author, and the worke in hand Geometry, it must needs be confest we are beholden to Euclides Elements for: And he that would be rich in that profession, may have, if he be not covetous, his fill there, if he will labour hard, and take paines for it, it is true. But in what time thinke yau, may a man learne all Euclide, and so by him be made skilfull in this Art? By himselfe I know not whether ever or never: And with the helpe of another, although very expert, I will not promise him that hee shall attaine to perfection in many yeares.

Hippocrates the Prince of Physicians hath, as they say, in his workes laid out the whole Art of Physicke; but I marvell how long a man should study him alone, and read him over and over, before he should be a good Physician? I feare mee all the friends that he hath, and neighbours round about him, yea, and himselfe too, would all die before he should be able to hele them, or per adventure ere he should be able to know what they ail'd; and after 30, or 40. yeeres of such his study, I would be very loath to commit my selfe unto him. How much therefore are the students of this noble Science beholding unto those men, who by their industry, practise, and painefull travells, have shewed them a ready and certaine way through this wildernesse?

The Elements of Euclide they do containe generally the whole art of Geometry: But if you will offer to travell thorow them alone, you shall finde them, I will warrant you, Elements indeed: for there you may walke through the spacious Aire, and over the great and wide sea, and in and about the vaste and arid wildernesse many a day and night, before you shall know where you are. This Ramus, my Authour in reading him found to be true; and confesseth himselfe often to have beene at a stand: Often to have lost himselfe: Often to have hitte upon a rocke, when he had thought he had touch'd land.

Least therefore other men, in this journey doe not likewise loose themselves, for the benefit and safety, I meane, of others he hath prick'd them out a charde or chack'd out a way, which if thou shalt please to follow, it shall lead thee to thy wayes end, as directly, and in as short time, as conveniently may be. Yet in what time I cannot warrant thee: For all mens capacity, especially in these Arts, is not alike: All are not a like painefull, industrious, or diligent: All are not of the same ability of body, to be able to continue or sit at it: Or all not so free from other imployments or businesse calling them from their study, as some others are. For know this for certaine, Thou shalt here make no great progresse, except thou doe make it as it were a continued labour, Here you must observe that rule of the great Painter, Nulla dies sine linea, Let no day passe over your head, in which you draw not some diagram or figure or other.

One other thing let me also advise thee of, how capable soever thou art, refuse not, if thou maist have it, the helpe of a teacher; For except thou be another Hippocrates or Forcatelus, whō our Authour mentioneth, thou canst not in these Arts and Sciences attaine unto any great perfection without infinite patience and great losse of most precious time, For they are therefore called Μαθηματικόι, Mathematicks, that is, doctrinal or disciplinary Arts, because they are not to be attained unto by our owne information and industry; but by the helpe and instruction of others.

This Worke gentle Reader, was in part above 30. yeares since published by M. Thomas Hood, a learned man, and loving friend of mine, who teaching these Arts, in the Staplers Chappell in Leadenhall London, for the benefit of his Schollers and Auditory, did set out the Elements apart by themselves. The whole at large, with the Diagrammes, and Demonstrations, hee promised, as appeareth in the Preface to that his Worke, at his convenient leysure to send out shortly, after them. This for ought we know or can learne, is not by him or any other performed: And yet are those alone, without these of small use or none to a learner, where a teacher is not alwayes at hand. Wherefore we are bold being (encouraged thereunto by some private friends, and especially by the learned M. H. Brigges, professour of Geometry in the famous Vniversity of Oxford) to publish this of ours long since finished and ended.

The usuall termes, whether Latine or Greeke, commonly used by the Geometers, we have set downe and expressed in English, as well as we could, as others, writing of this argument in our language, have done before us. These termes, I doubt not, may by some in English otherwise be expressed, but how harsh those termes, may unto Mathematicall eares, at the first appeare, I will not say; and use in short time will make these familiar, and as pleasing to the eare as those possibly may be.

Our Authour, in the declaration of the Elements hath many passages, which in our judgement doe not make so much for the understanding of the matter in hand, as for the defence of the method here used, against Aristotle, Euclide, Proclus, and others, which we have therfore wholly omitted. Some other things, which in our opinion, might in some respect illustrate any particular in this businesse, we have here and there inserted. Out of the learned Finkius's Geometria Rotundi, Wee have added to the fifth Booke certaine Propositions with their Consectaries out of Ptolomi's Almagest. The painfull and diligent Rod. Snellius out of the Lectures and Annotations of B. Salignacus, I. Tho. Freigius, and others, hath illustrated and altered here and there some few things.

The Contents.

Booke I. Of a Magnitude. Page [1]

Booke II. Of a Line. p. [13]

Book III. Of an Angle. p. [21]

Book IV. Of a Figure. p. [32]

Book V. Of Lines and Angles in a plaine Surface. p. [51]

Book VI. Of a Triangle. p. [83]

Book VII. The comparison of Triangles. p. [94]

Book VIII. Of the diverse kinds of Triangles. p. [106]

Book IX. Of the measuring of right lines by like right-angled Triangles. p. [113]

Book X. Of a Triangulate and Parallelogramme. p. [136]

Book XI. Of a Right-angle. p. [148]

Book XII. Of a Quadrate. p. [152]

Book XIII. Of an Oblong. p. [167]

Book XIV. Of a right line proportionally cut: And of other Quadrangles, and Multangles. p. [174]

Book XV. Of the Lines in a Circle. p. [201]*

Book XVI. Of the Segments of a Circle. p. [201]

Book XVII. Of the Adscription of a Circle and Triangle. p. [215]

Book XVIII. Of the adscription of a Triangulate. p. [221]

Book XIX. Of the measuring of ordinate Multangle, and of a Circle. p. [252]*

Book XX. Of a Bossed surface. p. [257]*

Book XXI. Of Lines and Surfaces in solids. p. [242]

Book XXII. Of a Pyramis. p. [249]

Book XXIII. Of a Prisma. p. [256]

Book XXIV. Of a Cube. p. [264]

Book XXV. Of mingled ordinate Polyedra's. p. [271]

Book XXVI. Of a Spheare. p. [284]

Book XXVII. Of the Cone and Cylinder. p. [290]

VIA REGIA AD GEOMETRIAM.

THE FIRST BOOKE OF Peter Ramus's Geometry, Which is of a Magnitude.

1. Geometry is the Art of measuring well.

The end or scope of Geometry is to measure well: Therefore it is defined of the end, as generally all other Arts are. To measure well therefore is to consider the nature and affections of every thing that is to be measured: To compare such like things one with another: And to understand their reason and proportion and similitude. For all that is to measure well, whether it bee that by Congruency and application of some assigned measure: Or by Multiplication of the termes or bounds: Or by Division of the product made by multiplication: Or by any other way whatsoever the affection of the thing to be measured be considered.

But this end of Geometry will appeare much more beautifull and glorious in the use and geometricall workes and

practise then by precepts, when thou shalt observe Astronomers, Geographers, Land-meaters, Sea-men, Enginers, Architects, Carpenters, Painters, and Carvers, in the description and measuring of the Starres, Countries, Lands, Engins, Seas, Buildings, Pictures, and Statues or Images to use the helpe of no other art but of Geometry. Wherefore here the name of this art commeth farre short of the thing meant by it. (For Geometria, made of Gè, which in the Greeke language signifieth the Earth; and Métron, a measure, importeth no more, but as one would say Land-measuring. And Geometra, is but Agrimensor, A land-meter: or as Tully calleth him Decempedator, a Pole-man: or as Plautus, Finitor, a Marke-man.) when as this Art teacheth not only how to measure the Land or the Earth, but the Water, and the Aire, yea and the whole World too, and in it all Bodies, Surfaces, Lines, and whatsoever else is to bee measured.

Now a Measure, as Aristotle doth determine it, in every thing to be measured, is some small thing conceived and set out by the measurer; and of the Geometers it is called Mensura famosa, a knowne measure. Which kinde of measures, were at first, as Vitruvius and Herodo teache us, taken from mans body: whereupon Protagoras sayd, That man was the measure of all things, which speech of his, Saint Iohn, Apoc. 21. 17. doth seeme to approve. True it is, that beside those, there are some other sorts of measures, especially greater ones, taken from other things, yet all of them generally made and defined by those. And because the stature and bignesse of men is greater in some places, then it is ordinarily in others, therefore the measures taken from them are greater in some countries, then they are in others. Behold here a catalogue, and description of such as are commonly either used amongst us, or some times mentioned in our stories and other bookes translated into our English tongue.

Granum hordei, a Barley corne, like as a wheat corne in weights, is no kinde of measure, but is quiddam minimum

in mensura, some least thing in a measure, whereof it is, as it were, made, and whereby it is rectified.

Digitus, a Finger breadth, conteineth 2. barly cornes length, or foure layd side to side:

Pollex, a Thumbe breadth; called otherwise Vncia, an ynch, 3. barley cornes in length:

Palmus, or Palmus minor, an Handbreadth, 4. fingers, or 3. ynches.

Spithama, or Palmus major, a Span, 3. hands breadth, or 9. ynches.

Cubitus, a Cubit, halfe a yard, from the elbow to the top of the middle finger, 6. hands breadth, or two spannes.

Ulna, from the top of the shoulder or arme-hole, to the top of the middle finger. It is two folde; A yard and an Elne. A yard, containeth 2. cubites, or 3. foote: An Elne, one yard and a quarter, or 2. cubites and ½.

Pes, a Foot, 4. hands breadth, or twelve ynches.

Gradus, or Passus minor, a Steppe, two foote and an halfe.

Passus, or Passus major, a Stride, two steppes, or five foote.

Pertica, a Pertch, Pole, Rod or Lugge, 5. yardes and an halfe.

Stadium, a Furlong; after the Romans, 125. pases: the English, 40. rod.

Milliare, or Milliarium, that is mille passus, 1000. passes, or 8. furlongs.

Leuca, a League, 2. miles: used by the French, spaniards, and seamen.

Parasanga, about 4. miles: a Persian, & common Dutch mile; 30. furlongs.

Schœnos, 40. furlongs: an Egyptian, or swedland mile.

Now for a confirmation of that which hath beene saide, heare the words of the Statute.

It is ordained, That 3. graines of Barley, dry and round, do make an Ynch: 12. ynches do make a Foote: 3. foote do make a

Yard: 5. yardes and ½ doe make a Perch: And 40. perches in length, and 4. in breadth, doe make an Aker: 33. Edwar. 1. De terris mensurandis: & De compositione ulnarum & Perticarum.

Item, Bee it enacted by the authority aforesaid; That a Mile shall be taken and reckoned in this manner, and no otherwise; That is to say, a Mile to containe 8. furlongs: And every Furlong to containe 40. lugges or poles: And every Lugge or Pole to containe 16. foote and ½. 25. Eliza. An Act for restraint of new building, &c.

These, as I said, are according to diverse countries, where they are used, much different one from another: which difference, in my judgment; ariseth especially out of the difference of the Foote, by which generally they are all made, whether they be greater of lesser. For the Hand being as before hath beene taught, the fourth part of the foot whether greater or lesser: And the Ynch, the third part of the hand, whether greater or lesser.

Item, the Yard, containing 3. foote, whether greater or lesser: And the Rodde 5. yardes and ½, whether greater or lesser, and so forth of the rest; It must needes follow, that the Foote beeing in some places greater then it is in other some, these measures, the Hand, I meane, the Ynch, the Yard, the Rod, must needes be greater or lesser in some places then they are in other. Of this diversity therefore, and difference of the foot, in forreine countries, as farre as mine intelligence will informe me, because the place doth invite me, I will here adde these few lines following. For of the rest, because they are of more speciall use, I will God willing, as just occasion shall be administred, speake more plentifully hereafter.

Of this argument divers men have written somewhat, more or lesse: But none to my knowledge, more copiously and curiously, then Iames Capell, a Frenchman, and the learned Willebrand, Snellius, of Leiden in Holland, for they have compared, and that very diligently, many and sundry kinds of these measures one with another. The first as you may

see in his treatise De mensuris intervallorum describeth these eleven following: of which the greatest is Pes Babylonius, the Babylonian foote; the least, Pes Toletanus, the foote used about Toledo in Spaine: And the meane betweene both, Pes Atticus, that used about Athens in Greece. For they are one unto another as 20. 15. and 12. are one unto another. Therefore if the Spanish foote, being the least, be devided into 12. ynches, and every inch againe into 10. partes, and so the whole foote into 120. the Atticke foote shall containe of those parts 150. and the Babylonian, 200. To this Atticke foote, of all other, doth ours come the neerest: For our English foote comprehendeth almost 152. such parts.

The other, to witt the learned Snellius, in his Eratosthenes Batavus, a booke which hee hath written of the true quantity of the compasse of the Earth, describeth many more, and that after a farre more exact and curious manner.

Here observe, that besides those by us here set downe, there are certaine others by him mentioned, which as hee writeth are found wholly to agree with some one or other of these. For Rheinlandicus, that of Rheinland or Leiden, which hee maketh his base, is all one with Romanus, the Italian or Roman foote. Lovaniensis, that of Lovane, with that of Antwerpe: Bremensis, that of Breme in Germany, with that of Hafnia, in Denmarke. Onely his Pes Arabicus, the Arabian foote, or that mentioned in Abulfada, and Nubiensis: the Geographers I have overpassed, because hee dareth not, for certeine, affirme what it was.

Looke of what parts Pes Tolitanus, the spanish foote, or that of Toledo in Spaine, conteineth 120. of such is the Pes.

Heidelbergicus, that of Heidelberg, 137.

Hetruscus, that of Tuscan, in Italie, 138.

Sedanensis, of Sedan in France, 139.

Romanus, that of Rome in Italy, 144.

Atticus, of Athens in Greece, 150.

Anglicus, of England, 152.

Parisinus, of Paris in France, 160.

Syriacus, of Syria, 166.

Ægyptiacus, of Egypt, 171.

Hebraicus, that of Iudæa, 180.

Babylonius, that of Babylon, 200.

Looke of what parts Pes Romanus, the foote of Rome, (which is all one with the foote of Rheinland) is 1000. of such parts is the foote of

Toledo, in Spaine, 864.

Mechlin, in Brabant, 890.

Strausburgh, in Germany, 891.

Amsterdam, in Holland, 904.

Antwerpe, in Brabant, 909.

Bavaria, in Germany, 924.

Coppen-haun, in Denmarke, 934.

Goes, in Zeland, 954.

Middleburge, in Zeland, 960.

London, in England, 968.

Noremberge, in Germany, 974.

Ziriczee, in Zeland, 980.

The ancient Greeke, 1042.

Dort, in Holland, 1050.

Paris, in France, 1055.

Briel, in Holland, 1060.

Venice, in Italy, 1101.

Babylon, in Chaldæa, 1172.

Alexandria, in Egypt, 1200.

Antioch, in Syria, 1360.

Of all other therefore our English foote commeth neerest unto that used by the Greekes: And the learned Master Ro. Hues, was not much amisse, who in his booke or Treatise De Globis, thus writeth of it Pedem nostrum Angli cum Græcorum pedi æqualem invenimus, comparatione facta

cum Græcorum pede, quem Agricola & alij ex antiquis monumentis tradiderunt.

Now by any one of these knowne and compared with ours, to all English men well knowne the rest may easily be proportioned out.

2. The thing proposed to bee measured is a Magnitude.

Magnitudo, a Magnitude or Bignesse is the subject about which Geometry is busied. For every Art hath a proper subject about which it doth employ al his rules and precepts: And by this especially they doe differ one from another. So the subject of Grammar was speech; of Logicke, reason; of Arithmeticke, numbers; and so now of Geometry it is a magnitude, all whose kindes, differences and affections, are hereafter to be declared.

3. A Magnitude is a continuall quantity.

A Magnitude is quantitas continua, a continued, or continuall quantity. A number is quantitas discreta, a disjoined quantity: As one, two, three, foure; doe consist of one, two, three, foure unities, which are disjoyned and severed parts: whereas the parts of a Line, Surface, and Body are contained and continued without any manner of disjunction, separation, or distinction at all, as by and by shall better and more plainely appeare. Therefore a Magnitude is here understood to be that whereby every thing to be measured is said to bee great: As a Line from hence is said to be long, a Surface broade, a Body solid: Wherefore Length, Breadth, and solidity are Magnitudes.

4. That is continuum, continuall, whose parts are contained or held together by some common bound.

This definition of it selfe is somewhat obscure, and to be

understand onely in a geometricall sense: And it dependeth especially of the common bounde. For the parts (which here are so called) are nothing in the whole, but in a potentia or powre: Neither indeede may the whole magnitude bee conceived, but as it is compact of his parts, which notwithstanding wee may in all places assume or take as conteined and continued with a common bound, which Aristotle nameth a Common limit; but Euclide a Common section, as in a line, is a Point, in a surface, a Line: in a body, a Surface.

5. A bound is the outmost of a Magnitude.

Terminus, a Terme, or Bound is here understood to bee that which doth either bound, limite, or end actu, in deede; as in the beginning and end of a magnitude: Or potentia, in powre or ability, as when it is the common bound of the continuall magnitude. Neither is the Bound a parte of the bounded magnitude: For the thing bounding is one thing, and the thing bounded is another: For the Bound is one distance, dimension, or degree, inferiour to the thing bounded: A Point is the bound of a line, and it is lesse then a line by one degree, because it cannot bee divided, which a line may. A Line is the bound of a surface, and it is also lesse then a surface by one distance or dimension, because it is only length, wheras a surface hath both length and breadth. A Surface is the bound of a body, and it is lesse likewise then it is by one dimension, because it is onely length and breadth, whereas as a body hath both length, breadth, and thickenesse.

Now every Magnitude actu, in deede, is terminate, bounded and finite, yet the geometer doth desire some time to have an infinite line granted him, but no otherwise infinite or farther to bee drawane out then may serve his turne.

6. A Magnitude is both infinitely made, and continued, and cut or divided by those things wherewith it is bounded.

A line, a surface, and a body are made gemetrically by the motion of a point, line, and surface: Item, they are conteined, continued, and cut or divided by a point, line, and surface. But a Line is bounded by a point: a surface, by a line: And a Body by a surface, as afterward by their severall kindes shall be understood.

Now that all magnitudes are cut or divided by the same wherewith they are bounded, is conceived out of the definition of Continuum, e. 4. For if the common band to containe and couple together the parts of a Line, surface, & Body, be a Point, Line, and Surface, it must needes bee that a section or division shall be made by those common bandes: And that to bee dissolved which they did containe and knitt together.

7. A point is an undivisible signe in a magnitude.

A Point, as here it is defined, is not naturall and to bee perceived by sense; Because sense onely perceiveth that which is a body; And if there be any thing lesse then other to be perceived by sense, that is called a Point. Wherefore a Point is no Magnitude: But it is onely that which in a Magnitude is conceived and imagined to bee undivisible. And although it be voide of all bignesse or Magnitude, yet is it the beginning of all magnitudes, the beginning I meane potentiâ, in powre.

8. Magnitudes commensurable, are those which one and the same measure doth measure: Contrariwise, Magnitudes incommensurable are those, which the same measure cannot measure. 1, 2. d. X.

Magnitudes compared betweene themselves in respect of numbers have Symmetry or commensurability, and

Reason or rationality: Of themselves, Congruity and Adscription. But the measure of a magnitude is onely by supposition, and at the discretion of the Geometer, to take as pleaseth him, whether an ynch, an hand breadth, foote, or any other thing whatsoever, for a measure. Therefore two magnitudes, the one a foote long, the other two foote long, are commensurable; because the magnitude of one foote doth measure them both, the first once, the second twice. But some magnitudes there are which have no common measure, as the Diagony of a quadrate and his side, 116. p. X. actu, in deede, are Asymmetra, incommensurable: And yet they are potentiâ, by power, symmetra, commensurable, to witt by their quadrates: For the quadrate of the diagony is double to the quadrate of the side.

9. Rationall Magnitudes are those whose reason may bee expressed by a number of the measure given. Contrariwise they are irrationalls. 5. d. X.

Ratio, Reason, Rate, or Rationality, what it is our Authour (and likewise Salignacus) have taught us in the first Chapter of the second booke of their Arithmetickes: Thither therefore I referre thee.

Data mensura, a Measure given or assigned, is of Euclide called Rhetè, that is spoken, (or which may be uttered) definite, certaine, to witt which may bee expressed by some number, which is no other then that, which as we said, was called mensura famosa, a knowne or famous measure.

Therefore Irrationall magnitudes, on the contrary, are understood to be such whose reason or rate may not bee expressed by a number or a measure assigned: As the side of the side of a quadrate of 20. foote unto a magnitude of two foote; of which kinde of magnitudes, thirteene sorts are mentioned in the tenth booke of Euclides Elements: such are the segments of a right line proportionally cutte, unto the whole line. The Diameter in a circle is rationall:

But it is irrationall unto the side of an inscribed quinquangle: The Diagony of an Icosahedron and Dodecahedron is irrationall unto the side.

10. Congruall or agreeable magnitudes are those, whose parts beeing applyed or laid one upon another doe fill an equall place.

Symmetria, Symmetry or Commensurability and Rate were from numbers: The next affections of Magnitudes are altogether geometricall.

Congruentia, Congruency, Agreeablenesse is of two magnitudes, when the first parts of the one doe agree to the first parts of the other, the meane to the meane, the extreames or ends to the ends, and lastly the parts of the one, in all respects to the parts, of the other: so Lines are congruall or agreeable, when the bounding, points of the one, applyed to the bounding points of the other, and the whole lengths to the whole lengthes, doe occupie or fill the same place. So Surfaces doe agree, when the bounding lines, with the bounding lines: And the plots bounded, with the plots bounded doe occupie the same place. Now bodies if they do agree, they do seeme only to agree by their surfaces. And by this kind of congruency do we measure the bodies of all both liquid and dry things, to witt, by filling an equall place. Thus also doe the moniers judge the monies and coines to be equall, by the equall weight of the plates in filling up of an equall place. But here note, that there is nothing that is onely a line, or a surface onely, that is naturall and sensible to the touch, but whatsoever is naturall, and thus to be discerned is corporeall.

Therefore

11. Congruall or agreeable Magnitudes are equall. 8. ax. j.

A lesser right line may agree to a part of a greater, but to so much of it, it is equall, with how much it doth agree:

Neither is that axiome reciprocall or to be converted: For neither in deede are Congruity and Equality reciprocall or convertible. For a Triangle may bee equall to a Parallelogramme, yet it cannot in all points agree to it: And so to a Circle there is sometimes sought an equall quadrate, although incongruall or not agreeing with it: Because those things which are of the like kinde doe onely agree.

12. Magnitudes are described betweene themselves, one with another, when the bounds of the one are bounded within the boundes of the other: That which is within, is called the inscript: and that which is without, the Circumscript.

Now followeth Adscription, whose kindes are Inscription and Circumscription; That is when one figure is written or made within another: This when it is written or made about another figure.

Homogenea, Homogenealls or figures of the same kinde onely betweene themselves rectitermina, or right bounded, are properly adscribed betweene themselves, and with a round. Notwithstanding, at the 15. booke of Euclides Elements Heterogenea, Heterogenealls or figures of divers kindes are also adscribed, to witt the five ordinate plaine bodies betweene themselves: And a right line is inscribed within a periphery and a triangle.

But the use of adscription of a rectilineall and circle, shall hereafter manifest singular and notable mysteries by the reason and meanes of adscripts; which adscription shall be the key whereby a way is opened unto that most excellent doctrine taught by the subtenses or inscripts of a circle as Ptolomey speakes, or Sines, as the latter writers call them.

The second Booke of Geometry. Of a Line.

1. A Magnitude is either a Line or a Lineate.

The Common affections of a magnitude are hitherto declared: The Species or kindes doe follow: for other then this division our authour could not then meete withall.

2. A Line is a Magnitude onely long.

As are ae. io. and uy. such a like Magnitude is conceived in the measuring of waies, or distance of one place from another: And by the difference of a lightsome place from a darke: Euclide at the 2 d j. defineth a line to be a length void of breadth: And indeede length is the proper difference of a line, as breadth is of a face, and solidity of a body.

3. The bound of a line is a point.

Euclide at the 3. d j. saith that the extremities or ends of a line are points. Now seeing that a Periphery or an hoope line hath neither beginning nor ending, it seemeth not to bee bounded with points: But when it is described or made it beginneth at a point, and it endeth at a pointe. Wherefore a Point is the bound of a line, sometime actu, in deed, as in a right line: sometime potentiâ, in a possibility, as in a perfect periphery. Yea in very deede, as before was taught in the definition of continuum, 4 e. all lines, whether they bee right lines, or crooked, are contained or continued with points. But a line is made by the

motion of a point. For every magnitude generally is made by a geometricall motion, as was even now taught, and it shall afterward by the severall kindes appeare, how by one motion whole figures are made: How by a conversion, a Circle, Spheare, Cone, and Cylinder: How by multiplication of the base and heighth, rightangled parallelogrammes are made.

4. A Line is either Right or Crooked.

This division is taken out of the 4 d j. of Euclide, where rectitude or straightnes is attributed to a line, as if from it both surfaces and bodies were to have it. And even so the rectitude of a solid figure, here-after shall be understood by a right line perpendicular from the toppe unto the center of the base. Wherefore rectitude is propper unto a line: And therefore also obliquity or crookednesse, from whence a surface is judged to be right or oblique, and a body right or oblique.

5. A right line is that which lyeth equally betweene his owne bounds: A crooked line lieth contrariwise. 4. d. j.

Now a line lyeth equally betweene his owne bounds, when it is not here lower, nor there higher: But is equall to the space comprehended betweene the two bounds or ends: As here ae. is, so hee that maketh rectum iter, a journey in a straight line, commonly he is said to treade so much ground, as he needes must, and no more: He goeth obliquum iter, a crooked way, which goeth more then he needeth, as Proclus saith.

6. A right line is the shortest betweene the same bounds.

Linea recta, a straight or right line is that, as Plato defineth it, whose middle points do hinder us from seeing both the extremes at once; As in the eclipse of the Sunne, if a right line should be drawne from the Sunne, by the Moone, unto our eye, the body of the Moone beeing in the midst, would hinder our sight, and would take away the sight of the Sunne from us: which is taken from the Opticks, in which we are taught, that we see by straight beames or rayes. Therfore to lye equally betweene the boundes, that is by an equall distance: to bee the shortest betweene the same bounds; And that the middest doth hinder the sight of the extremes, is all one.

7. A crooked line is touch'd of a right or crooked line, when they both doe so meete, that being continued or drawne out farther they doe not cut one another.

Tactus, Touching is propper to a crooked line, compared either with a right line or crooked, as is manifest out of the 2. and 3. d 3. A right line is said to touch a circle, which touching the circle and drawne out farther, doth not cut the circle, 2 d 3. as here ae, the right line toucheth the periphery iou. And ae. doth touch the helix or spirall.

Circles are said to touch one another, when touching they doe not cutte one another, 3. d 3. as here the periphery doth aej. doth touch the periphery ouy.

Therefore

8. Touching is but in one point onely. è 13. p 3.

This Consectary is immediatly conceived out of the definition; for otherwise it were a cutting, not touching. So Aristotle in his Mechanickes saith; That a round is easiliest mou'd and most swift; Because it is least touch't of the plaine underneath it.

9. A crooked line is either a Periphery or an Helix. This also is such a division, as our Authour could then hitte on.

10. A Periphery is a crooked line, which is equally distant from the middest of the space comprehended.

Peripheria, a Periphery, or Circumference, as eio. doth stand equally distant from a, the middest of the space enclosed or conteined within it.

Therefore

11. A Periphery is made by the turning about of a line, the one end thereof standing still, and the other drawing the line.

As in eio. let the point a stand still: And let the line ao, be turned about, so that the point o doe make a race, and it shall make the periphery eoi. Out of this fabricke doth Euclide, at the 15. d. j. frame the definition of a Periphery: And so doth hee afterwarde define a Cone, a Spheare, and a Cylinder.

Now the line that is turned about, may in a plaine, bee either a right line or a crooked line: In a sphericall it is onely a crooked line; But in a conicall or Cylindraceall it may bee a right line, as is the side of a Cone and Cylinder. Therefore in the conversion or turning about of a line making a periphery, there is considered onely the distance; yea two points, one in the center, the other in the toppe, which therefore Aristotle nameth Rotundi principia, the principles or beginnings of a round.

12. An Helix is a crooked line which is unequally distant from the middest of the space, howsoever inclosed.

Hæc tortuosa linea, This crankled line is of Proclus called Helicoides. But it may also be called Helix, a twist or wreath: The Greekes by this word do commonly either understand one of the kindes of Ivie which windeth it selfe about trees & other plants; or the strings of the vine, whereby it catcheth hold and twisteth it selfe about such things as are set for it to clime or run upon. Therfore it should properly signifie the spirall line. But as it is here taken it hath divers kindes; As is the Arithmetica which is Archimede'es Helix, as the Conchois, Cockleshell-like: as is the Cittois, Iuylike: The Tetragonisousa, the Circle squaring line, to witt that by whose meanes a circle may be brought into a square: The Admirable line, found out by Menelaus: The Conicall Ellipsis, the Hyperbole, the Parabole, such as these are, they attribute to

Menechmus: All these Apollonius hath comprised in eight Bookes; but being mingled lines, and so not easie to bee all reckoned up and expressed, Euclide hath wholly omitted them, saith Proclus, at the 9. p. j.

13. Lines are right one unto another, whereof the one falling upon the other, lyeth equally: Contrariwise they are oblique. è 10. d j.

Hitherto straightnesse and crookednesse have beene the affections of one sole line onely: The affections of two lines compared one with another are Perpendiculum, Perpendicularity and Parallelismus, Parallell equality; Which affections are common both to right and crooked lines. Perpendicularity is first generally defined thus:

Lines are right betweene themselves, that is, perpendicular one unto another, when the one of them lighting upon the other, standeth upright and inclineth or leaneth neither way. So two right lines in a plaine may bee perpendicular; as are ae. and io. so two peripheries upon a sphearicall may be perpendiculars, when the one of them falling upon the other, standeth indifferently betweene, and doth not incline or leane either way. So a right line may be

perpendicular unto a periphery, if falling upon it, it doe reele neither way, but doe ly indifferently betweene either side. And in deede in all respects lines right betweene themselves, and perpendicular lines are one and the same. And from the perpendicularity of lines, the perpendicularity of surfaces is taken, as hereafter shall appeare. Of the perpendicularity of bodies, Euclide speaketh not one word in his Elements, & yet a body is judged to be right, that is, plumme or perpendicular unto another body, by a perpendicular line.

Therefore,

14. If a right line be perpendicular unto a right line, it is from the same bound, and on the same side, one onely. ê 13. p. xj.

Or, there can no more fall from the same point, and on the same side but that one. This consectary followeth immediately upon the former: For if there should any more fall unto the same point and on the same side, one must needes reele, and would not ly indifferently betweene the parts cut: as here thou seest in the right line ae. io. eu.

15. Parallell lines they are, which are everywhere equally distant. è 35. d j.

Parallelismus, Parallell-equality doth now follow: And this also is common to crooked lines and right lines: As

heere thou seest in these examples following.

Parallell-equality is derived from perpendicularity, and is of neere affinity to it. Therefore Posidonius did define it by a common perpendicle or plum-line: yea and in deed our definition intimateth asmuch. Parallell-equality of bodies is no where mentioned in Euclides Elements: and yet they may also bee parallells, and are often used in the Optickes, Mechanickes, Painting and Architecture.

Therefore,

16. Lines which are parallell to one and the same line, are also parallell one to another.

This element is specially propounded and spoken of right lines onely, and is demonstrated at the 30. p. j. But by an addition of equall distances, an equall distance is knowne, as here.

The third Booke of Geometry. Of an Angle.

1. A lineate is a Magnitude more then long.

A New forme of doctrine hath forced our Authour to use oft times new words, especially in dividing, that the logicall lawes and rules of more perfect division by a dichotomy, that is into two kindes, might bee held and observed. Therefore a Magnitude was divided into two kindes, to witt into a Line and a Lineate: And a Lineate is made the genus of a surface and a Body. Hitherto a Line, which of all bignesses is the first and most simple, hath been described: Now followeth a Lineate, the other kinde of magnitude opposed as you see to a line, followeth next in order. Lineatum therefore a Lineate, or Lineamentum, a Lineament, (as by the authority of our Authour himselfe, the learned Bernhard Salignacus, who was his Scholler, hath corrected it) is that Magnitude in which there are lines: Or which is made of lines, or as our Authour here, which is more then long: Therefore lines may be drawne in a surface, which is the proper soile or plots of lines; They may also be drawne in a body, as the Diameter in a Prisma: the axis in a spheare; and generally all lines falling from aloft: And therfore Proclus maketh some plaine, other solid lines. So Conicall lines, as the Ellipsis, Hyperbole, and Parabole, are called solid lines because they do arise from the cutting of a body.

2. To a Lineate belongeth an Angle and a Figure.

The common affections of a Magnitude were to be bounded, cutt, jointly measured, and adscribed: Then of a line to be right, crooked, touch'd, turn'd about, and

wreathed: All which are in a lineate by meanes of a line. Now the common affections of a Lineate are to bee Angled and Figured. And surely an Angle and a figure in all Geometricall businesses doe fill almost both sides of the leafe. And therefore both of them are diligently to be considered.

3. An Angle is a lineate in the common section of the bounds.

So Angulus Superficiarius, a superficiall Angle, is a surface consisting in the common section of two lines: So angulus solidus, a solid angle, in the common section of three surfaces at the least.

[But the learned B. Salignacus hath observed, that all angles doe not consist in the common section of the bounds, Because the touching of circles, either one another, or a rectilineal surface doth make an angle without any cutting of the bounds: And therefore he defineth it thus: Angulus est terminorum inter se invicem inclinantium concursus: An angle is the meeting of bounds, one leaning towards another.] So is aei. a superficiall angle: [And such also are the angles ouy. and bcd.] so is the angle o. a solid angle, to witt comprehended of the three surfaces aoi. ioe. and aoe. Neither may a surface, of 2. dimensions, be bounded with

one right line: Nor a body, of three dimensions, bee bounded with two, at lest beeing plaine surfaces.

4. The shankes of an angle are the bounds compreding the angle.

Scèle or Crura, the Shankes, Legges, H. are the bounds insisting or standing upon the base of the angle, which in the Isosceles only or Equicrurall triangle are so named of Euclide, otherwise he nameth them Latera, sides. So in the examples aforesaid, ea. and ei. are the shankes of the superficiary angle e; And so are the three surfaces aoi. ieo. and aeo. the shankes of the said angle o. Therefore the shankes making the angle are either Lines or Surfaces: And the lineates formed or made into Angles, are either Surfaces or Bodies.

5. Angles homogeneall, are angles of the same kinde, both in respect of their shankes, as also in the maner of meeting of the same: [Heterogeneall, are those which differ one from another in one, or both these conditions.]

Therefore this Homogenia, or similitude of angles is twofolde, the first is of shanks; the other is of the manner of meeting of the shankes: so rectilineall right angles, are angles homogeneall betweene themselves. But right-lined right angles, and oblique-lined right angles between themselves, are heterogenealls. So are neither all obtusangles compared to all obtusangles: Nor all acutangles, to all acutangles, homogenealls, except both these conditions doe concurre, to witt the similitude both of shanke and manner of meeting. Lunularis, a Lunular, or Moonlike corner angle is homogeneall to a Systroides and Pelecoides, Hatchet formelike, in shankes: For each of these are comprehended of

peripheries: The Lunular of one convexe; the other concave; as iue. The Systroides of both convex, as iao. The Pelecoides of both concave, as eau. And yet a lunular, in respect of the meeting of the shankes is both to the Systroides and Pelecoides heterogeneall: And therefore it is absolutely heterogeneall to it.

6. Angels congruall in shankes are equall.

This is drawne out of the [10. e j]. For if twice two shanks doe agree, they are not foure, but two shankes, neither are they two equall angles, but one angle. And this is that which Proclus speaketh of, at the 4. p j. when hee saith, that a right lined angle is equall to a right lined angle, when one of the shankes of the one put upon one of the shankes of the other, the other two doe agree: when that other shanke fall without, the angle of the out-falling shanke is the greater: when it falleth within, it is lesser: For there is comprehendeth; here it is comprehended.

Notwithstanding although congruall or agreeable angles be equall: yet are not congruity and equality reciprocall or convertible: For a Lunular may bee equall to a right

lined right angle, as here thou seest: For the angles of equall semicircles ieo. and aeu. are equall, as application doth shew. The angle aeo. is common both to the right angle aei. and to the lunar aueo. Let therefore the equall angle aeo. bee added to both: the right angle aei. shall be equall to the Lunular aueo.

The same Lunular also may bee equall to an obtusangle and Acutangle, as the same argument will demonstrate.

Therefore,

7. If an angle being equicrurall to an other angle, be also equall to it in base, it is equall: And if an angle having equall shankes with another, bee equall to it in the angle, it is also equall to it in the base. è 8. & 4. p j.

For such angles shall be congruall or agreeable in shanks, and also congruall in bases. Angulus isosceles, or Angulus æquicrurus, is a triangle having equall shankes unto another.

8. And if an angle equall in base to another, be also equall to it in shankes, it is equall to it.

For the congruency is the same: And yet if equall angles bee equall in base, they are not by and by equicrurall, as in the angles of the same section will appeare, as here. And so of two equalities, the first is reciprocall: The second is not. [And therefore is this Consectary, by the learned B. Salignacus, justly, according to the judgement of the worthy Rud. Snellius, here cancelled; or quite put out: For angles may be equall, although they bee unequall in shankes or in bases, as here, the angle a. is not greater then the angle o, although the angle o have both greater shankes and greater base then the angle a.]

And

9. If an angle equicrurall to another angle, be greater then it in base, it is greater: And if it be greater, it is greater in base: è 52 & 24. p j.

As here thou seest; [The angles eai. and uoy. are equicrurall, that is their shankes are equall one to another; But the base ei is greater then the base uy: Therefore the angle eai, is greater then the angle uoy. And contrary wise, they being equicrurall, and the angle eai. being greater then the angle uoy. The base ei. must needes be greater then the base uy.]

And

10. If an angle equall in base, be lesse in the inner shankes, it is greater.

Or as the learned Master T. Hood doth paraphrastically translate it. If being equall in the base, it bee lesser in the feete (the feete being conteined within the feete of the other angle) it is the greater angle. [That is, if one angle enscribed within another angle, be equall in base, the angle of the inscribed shall be greater then the angle of the circumscribed.]

As here the angle aoi. within the angle aei. And the bases are equall, to witt one and the same; Therefore aoi. the inner angle is greater then aei. the outter angle. Inner is added of necessity: For otherwise there will, in the section or cutting one of another, appeare a manifest errour. All these consectaries are drawne out of that same axiome of congruity, to witt out of the [10. e j]. as Proclus doth plainely affirme and teach: It seemeth saith hee, that the equalities of shankes and bases, doth cause the equality of the verticall angles. For neither, if the bases be equall, doth the equality of the shankes leave the same or equall angles: But if the base bee lesser, the angle decreaseth: If greater, it increaseth. Neither if the bases bee equall, and the shankes unequall, doth the angle remaine the same: But when they are made lesse, it is increased: when they are made greater, it is diminished: For the contrary falleth out to the angles and shankes of the angles. For if thou shalt imagine the shankes to be in the same base thrust downeward, thou makest them lesse, but their angle greater: but if thou do againe conceive them to be pul'd up higher, thou makest them greater, but their angle lesser. For looke how much more neere they come one to another, so much farther off is the toppe removed from the base: wherefore you may boldly affirme, that the same

base and equall shankes, doe define the equality of Angels. This Poclus,

Therefore,

11. If unto the shankes of an angle given, homogeneall shankes, from a point assigned, bee made equall upon an equall base, they shall comprehend an angle equall to the angle given. è 23. p j. & 26. p xj.

[This consectary teacheth how unto a point given, to make an angle equall to an Angle given. To the effecting and doing of each three things are required; First, that the shankes be homogeneall, that is in each place, either straight or crooked: Secondly, that the shankes bee made equall, that is of like or equall bignesse: Thirdly, that the bases be equall: which three conditions if they doe meete, it must needes be that both the angles shall bee equall: but if one of them be wanting, of necessity againe they must be unequall.]

This shall hereafter be declared and made plaine by many and sundry practises: and therefore here we bring no example of it.

12. An angle is either right or oblique.

Thus much of the Affections of an angle; the division into his kindes followeth. An angle is either Right or Oblique: as afore, at the 4 e ij. a line was right or straight, and oblique or crooked.

13. A right angle is an angle whose shankes are right (that is perpendicular) one unto another: An Oblique angle is contrary to this.

As here the angle aio. is a right angle, as is also oie. because the shanke oi. is right, that is, perpendicular to ae. [The instrument wherby they doe make triall which is a right angle, and which is oblique, that is greater or lesser then a right angle, is the square which carpenters and joyners do ordinarily use: For lengthes are tried, saith Vitruvius, by the Rular and Line: Heighths, by the Perpendicular or Plumbe: And Angles, by the square.] Contrariwise, an Oblique angle it is, when the one shanke standeth so upon another, that it inclineth, or leaneth more to one side, then it doth to the other: And one angle on the one side, is greater then that on the other.

Therefore,

14. All straight-shanked right angles are equall.

[That is, they are alike, and agreeable, or they doe fill the same place; as here are aio. and eio. And yet againe on the contrary: All straight shanked equall angles, are not right-angles.]

The axiomes of the equality of angles were three, as even now wee heard, one generall, and two Consectaries: Here moreover is there one speciall one of the equality of Right angles.

Angles therfore homogeneall and recticrurall, that is whose shankes are right, as are right lines, as plaine surfaces (For let us so take the word) are equall right

angles. So are the above written rectilineall right angles equall: so are plaine solid right angles, as in a cube, equall. The axiome may therefore generally be spoken of solid angles, so they be recticruralls: Because all semicircular right angles are not equall to all semicircular right angles: As here, when the diameter is continued it is perpendicular, and maketh twice two angles, within and without, the outter equall betweene themselves, and inner equall betweene themselves: But the outer unequall to the inner: And the angle of a greater semicircle is greater, then the angle of a lesser. Neither is this affection any way reciprocall, That all equall angles should bee right angles. For oblique angles may bee equall betweene themselves: And an oblique angle may bee made equall to a right angle, as a Lunular to a rectilineall right angle, as was manifest, at the [6 e].

The definition of an oblique is understood by the obliquity of the shankes: whereupon also it appeareth; That an oblique angle is unequall to an homogeneall right angle: Neither indeed may oblique angles be made equall by any lawe or rule: Because obliquity may infinitly bee both increased and diminished.

15. An oblique angle is either Obtuse or Acute.

One difference of Obliquity wee had before at the [9 e ij]. in a line, to witt of a periphery and an helix; Here there is another dichotomy of it into obtuse and acute: which difference is proper to angles, from whence it is translated or conferred upon other things and metaphorically used, as Ingenium obtusum, acutum; A dull, and quicke witte, and such like.

16. An obtuse angle is an oblique angle greater then a right angle. 11. d j.

Obtusus, Blunt or Dull; As here aei. In the definition the genus of both Species or kinds is to bee understood: For a right lined right angle is greater then a sphearicall right angle, and yet it is not an obtuse or blunt angle: And this greater inequality may infinitely be increased.

17. An acutangle is an oblique angle lesser then a right angle. 12. d j.

Acutus, Sharpe, Keene, as here aei. is. Here againe the same genus is to bee understood: because every angle which is lesse then any right angle is not an acute or sharp angle. For a semicircle and sphericall right angle, is lesse then a rectilineall right angle, and yet it is not an acute angle.

The fourth Booke, which is of a Figure.

1. A figure is a lineate bounded on all parts.

So the triangle aei. is a figure; Because it is a plaine bounded on all parts with three sides. So a circle is a figure: Because it is a plaine every way bounded with one periphery.

2. The center is the middle point in a figure.

In some part of a figure the Center, Perimeter, Radius, Diameter and Altitude are to be considered. The Center therefore is a point in the midst of the figure; so in the triangle, quadrate, and circle, the center is, aei.

Centrum gravitatis, the center of weight, in every plaine magnitude is said to bee that, by the which it is handled or held up parallell to the horizon: Or it is that point whereby the weight being suspended doth rest, when it is caried. Therefore if any plate should in all places be alike heavie, the center of magnitude and weight would be one and the same.

3. The perimeter is the compasse of the figure.

Or, the perimeter is that which incloseth the figure. This definition is nothing else but the interpretation of the Greeke word. Therefore the perimeter of a Triangle is one line made or compounded of three lines. So the perimeter of the triangle a, is eio. So the perimeter of the circle a is a periphery, as in eio. So the perimeter of a Cube is a surface, compounded of sixe surfaces: And the perimeter of a spheare is one whole sphæricall surface, as hereafter shall appeare.

4. The Radius is a right line drawne from the center to the perimeter.

Radius, the Ray, Beame, or Spoake, as of the sunne, and

cart wheele: As in the figures under written are ae, ai, ao. It is here taken for any distance from the center, whether they be equall or unequall.

5. The Diameter is a right line inscribed within the figure by his center.

As in the figure underwritten are ae, ai, ao. It is called the Diagonius, when it passeth from corner to corner. In solids it is called the Axis, as hereafter we shall heare.

Therefore,

6. The diameters in the same figure are infinite.

Although of an infinite number of unequall lines that be only the diameter, which passeth by or through the center

notwithstanding by the center there may be divers and sundry. In a circle the thing is most apparent: as in the Astrolabe the index may be put up and downe by all the points of the periphery. So in a speare and all rounds the thing is more easie to be conceived, where the diameters are equall: yet notwithstanding in other figures the thing is the same. Because the diameter is a right line inscribed by the center, whether from corner to corner, or side to side, the matter skilleth not. Therefore that there are in the same figure infinite diameters, it issueth out of the difinition of a diameter.

And

7. The center of the figure is in the diameter.

As here thou seest a, e, i this ariseth out of the definition of the diameter. For because the diameter is inscribed into the figure by the center: Therefore the Center of the figure must needes be in the diameter thereof: This is by Archimedes assumed especially at the 9, 10, 11, and 13 Theoreme of his Isorropicks, or Æquiponderants.

This consectary, saith the learned Rod. Snellius, is as it were a kinde of invention of the center. For where the diameters doe meete and cutt one another, there must the center needes bee. The cause of this is for that in every figure

there is but one center only: And all the diameters, as before was said, must needes passe by that center.

And

8. It is in the meeting of the diameters.

As in the examples following. This also followeth out of the same definition of the diameter. For seeing that every diameter passeth by the center: The center must needes be common to all the diameters: and therefore it must also needs be in the meeting of them: Otherwise there should be divers centers of one and the same figure. This also doth the same Archimedes propound in the same words in the 8. and 12 theoremes of the same booke, speaking of Parallelogrammes and Triangles.

9. The Altitude is a perpendicular line falling from the toppe of the figure to the base.

Altitudo, the altitude, or heigth, or the depth: [For that, as hereafter shall bee taught, is but Altitudo versa, an heighth

with the heeles upward.] As in the figures following are ae, io, uy, or sr. Neither is it any matter whether the base be the same with the figure, or be continued or drawne out longer, as in a blunt angled triangle, when the base is at the blunt corner, as here in the triangle, aei, is ao.

10. An ordinate figure, is a figure whose bounds are equall and angles equall.

In plaines the Equilater triangle is onely an ordinate figure, the rest are all inordinate: In quadrangles, the Quadrate is ordinate, all other of that sort are inordinate: In every sort of Multangles, or many cornered figures one may be an ordinate. In crooked lined figures the Circle is ordinate, because it is conteined with equall bounds, (one bound alwaies equall to it selfe being taken for infinite many,) because it is equiangled, seeing (although in deede there be in it no angle) the inclination notwithstanding is every where alike and equall, and as it were the angle of the perphery be alwaies alike unto it selfe: whereupon of Plato and Plutarch a circle is said to be Polygonia, a multangle; and of Aristotle Holegonia, a totangle, nothing else but one whole angle. In mingled-lined figures there is nothing that is ordinate: In

solid bodies, and pyramids the Tetrahedrum is ordinate: Of Prismas, the Cube: of Polyhedrum's, three onely are ordinate, the octahedrum, the Dodecahedrum, and the Icosahedrum. In oblique-lined bodies, the spheare is concluded to be ordinate, by the same argument that a circle was made to bee ordinate.

11. A prime or first figure, is a figure which cannot be divided into any other figures more simple then it selfe.

So in plaines the triangle is a prime figure, because it cannot be divided into any other more simple figure although it may be cut many waies: And in solids, the Pyramis is a first figure: Because it cannot be divided into a more simple solid figure, although it may be divided into an infinite sort of other figures: Of the Triangle all plaines are made; as of a Pyramis all bodies or solids are compounded; such are aei. and aeio.

12. A rationall figure is that which is comprehended of a base and height rationall betweene themselves.

So Euclide, at the 1. d. ij. saith, that a rightangled parallelogramme is comprehended of two right lines perpendicular one to another, videlicet one multiplied by the other. For Geometricall comprehension is sometimes as it were in numbers a multiplication: Therefore if yee shall grant the base and height to bee rationalls betweene themselves,

that their reason I meane may be expressed by a number of the assigned measure, then the numbers of their sides being multiplyed one by another, the bignesse of the figure shall be expressed. Therefore a Rationall figure is made by the multiplying of two rationall sides betweene themselves.

Therefore,

13. The number of a rationall figure, is called a Figurate number: And the numbers of which it is made, the Sides of the figurate.

As if a Right angled parallelogramme be comprehended of the base foure, and the height three, the Rationall made shall be 12. which wee here call the figurate: and 4. and 3. of which it was made, we name sides.

14. Isoperimetrall figures, are figures of equall perimeter.

This is nothing else but an interpretation of the Greeke word; So a triangle of 16. foote about, is a isoperimeter to a triangle 16. foote about, to a quadrate 16. foote about, and to a circle 16. foote about.

15. Of isoperimetralls homogenealls that which is most ordinate, is greatest: Of ordinate isoperimetralls heterogenealls, that is greatest, which hath most bounds.

So an equilater triangle shall bee greater then an isoperimeter inequilater triangle; and an equicrurall, greater then an unequicrurall: so in quadrangles, the quadrate is greater then that which is not a quadrate: so an oblong more ordinate, is greater then an oblong lesse ordinate. So of those figures which are heterogeneall ordinates, the quadrate is greater then the Triangle: And the Circle, then the Quadrate.

16. If prime figures be of equall height, they are in reason one unto another, as their bases are: And contrariwise.

The proportion of first figures is here twofold; the first is direct in those which are of equall height. In Arithmeticke we learned; That if one number doe multiply many numbers, the products shall be proportionall unto the numbers, multiplyed. From hence in rationall figures the content of those which are of equall height is to bee expressed by a number. As in two right angled parallelogrammes, let 4. the same height, multiply 2. and 3. the bases: The products 8. and 12. the parallelogrammes made, are directly proportionall unto the bases 2. and 3. Therefore as 2. is unto 3. so is 8. unto 12. The same shall afterward appeare in right Prismes and Cylinders. In plaines, Parallelogramms are the doubles of triangles: In solids, Prismes are the triples of pyramides: Cylinders, the triples of Cones. The converse of this element is plaine out of the former also: First figures if they be in reason one to another as their bases are, then are they of equall height, to witt when their products are proportionall unto the multiplyed, the same number did multiply them.

Therefore,

17. If prime figures of equall heighth have also equall bases, they are equall.

[The reason is, because then those two figures compared, have equall sides, which doe make them equall betweene themselves; For the parts of the one applyed or laid unto the parts of the other, doe fill an equall place, as was taught at the [10. e. j]. Sn.] So Triangles, so Parallelogrammes, and so other figures proposed are equalled upon an equall base.

18. If prime figures be reciprocall in base and height, they are equall: And contrariwise.

The second kind of proportion of first figures is reciprocall. This kinde of proportion rationall and expressible by a number, is not to be had in first figures themselves: but in those that are equally manifold to them, as was taught even now in direct proportion: As for example, Let these two right angled parallelogrammes, unequall in bases and heighths 3, 8, 4, 6, be as heere thou seest: The proportion reciprocall is thus, As 3 the base of the one, is unto 4, the base of the other: so is 6. the height of the one is to 8. the height of the other: And the parallelogrammes are equall, viz. 24. and 24. Againe, let two solids of unequall bases & heights (for here also the base is taken for the length and heighth) be 12, 2, 3, 6, 3, 4. The solids themselves shall be 72. and 72, as here thou seest; and the proportion of the bases and heights likewise is reciprocall: For as 24, is unto 18, so is 4, unto 3. The cause is out of the golden rule of proportion in Arithmeticke: Because twice two sides are

proportionall: Therefore the plots made of them shall be equall. And againe, by the same rule, because the plots are equall: Therefore the bounds are proportionall; which is the converse of this present element.

19. Like figures are equiangled figures, and proportionall in the shankes of the equall angles.

First like figures are defined, then are they compared one with another, similitude of figures is not onely of prime figures, and of such as are compounded of prime figures, but generally of all other whatsoever. This similitude consisteth in two things, to witt in the equality of their angles, and proportion of their shankes.

Therefore,

20. Like figures have answerable bounds subtended against their equall angles: and equall if they themselves be equall.

Or thus, They have their termes subtended to the equall angles correspondently proportionall: And equall if the figures themselves be equall; H. This is a consectary out of the former definition.

And

21. Like figures are situate alike, when the proportionall bounds doe answer one another in like situation.

The second consectary is of situation and place. And this like situation is then said to be when the upper parts of the one figure doe agree with the upper parts of the other, the lower, with the lower, and so the other differences of places. Sn.

And

22. Those figures that are like unto the same, are like betweene themselves.

This third consectary is manifest out of the definition of like figures. For the similitude of two figures doth conclude both the same equality in angles and proportion of sides betweene themselves.

And

23. If unto the parts of a figure given, like parts and alike situate, be placed upon a bound given, a like figure and likely situate unto the figure given, shall bee made accordingly.

This fourth consectary teacheth out of the said definition, the fabricke and manner of making of a figure alike and likely situate unto a figure given. Sn.

24. Like figures have a reason of their homologallor correspondent sides equally manifold unto their dimensions: and a meane proportionall lesse by one.

Plaine figures have but two dimensions, to witt Length, and Breadth: And therefore they have but a doubled reason of their homologall sides. Solids have three dimensions, videl. Length, Breadth, & thicknesse: therefore they shall have a treabled reason of their homologall or correspondent sides. In 8. and 18. the two plaines given, first the angles are equall: secondly, their homolegall side 2. and 4. and 3. and 6. are proportionall. Therefore the reason of 8. the first figure, unto 18. the

second, is as the reason is of 2. unto 3. doubled. But the reason of 2. unto 3. doubled, by the 3. chap. ij. of Arithmeticke, is of 4. to 9. (for 2/3 2/3 is 4/9.) Therefore the reason of 8. unto 18, that is, of the first figure unto the second, is of 4. unto 9. In Triangles, which are the halfes of rightangled parallelogrammes, there is the same truth, and yet by it selfe not rationall and to be expressed by numbers.

Said numbers are alike in the trebled reason of their homologall sides; As for example, 60. and 480. are like solids; and the solids also comprehended in those numbers are like-solids, as here thou seest: Because their sides, 4. 3. 5. and 8. 6. 10. are proportionall betweene themselves. But the reason of 60. to 480. is the reason of 4. to 8. trebled, thus 4/8 4/8 4/8 = 64/512; that is of 1. unto 8. or octupla, which you shall finde in the dividing of 480. by 60.

Thus farre of the first part of this element: The second, that like figurs have a meane, proportional lesse by one, then are their dimensions, shall be declared by few words. For plaines having but two dimensions, have but one meane proportionall, solids having three dimensions, have two meane proportionalls. The cause is onely Arithmeticall, as afore. For where the bounds are but 4. as they are in two plaines, there can be found no more but one meane proportionall, as in the former example of 8. and 18. where the homologall or correspondent sides are 2. 3. and 4. 6.

Therefore,

Againe by the same rule, where the bounds are 6. as they are in two solids, there may bee found no more but two meane proportionalls: as in the former solids 30. and 240. where the homologall or correspondent sides are 2. 4. 3. 6. 5. 10.

Therefore,

Therefore,

25. If right lines be continually proportionall, more by one then are the dimensions of like figures likelily situate unto the first and second, it shall be as the first right line is unto the last, so the first figure shall be unto the second: And contrariwise.

Out of the similitude of figures two consectaries doe arise, in part only, as is their axiome, rationall and expressable by numbers. If three right lines be continually proportionall, it shall be as the first is unto the third: So the rectilineall figure made upon the first, shall be unto the rectilineall figure made upon the second, alike and likelily situate. This may in some part be conceived and understood by numbers. As for example, Let the lines given, be 2. foot, 4. foote, and 8 foote. And upon the first and second, let there be made like figures, of 6. foote and 24. foote; So I meane, that 2. and 4. be the bases of them. Here as 2. the first line, is unto 8. the third line: So is 6. the first figure, unto 24. the second figure, as here thou seest.

Againe, let foure lines continually proportionall, be 1. 2. 4. 8. And let there bee two like solids made upon the first and second: upon the first, of the sides 1. 3. and 2. let it be 6. Upon the second, of the sides 2. 6. and 4. let it be 48. As the first right line 1. is unto the fourth 8. So is the figure 6. unto the second 48. as is manifest by division. The examples are thus.

Moreover by this Consectary a way is laid open leading unto the reason of doubling, treabling, or after any manner way whatsoever assigned increasing of a figure given. For as the first right line shall be unto the last: so shall the first figure be unto the second.

And

26. If foure right lines bee proportionall betweene themselves: Like figures likelily situate upon them, shall be also proportionall betweene themselves: And contrariwise, out of the 22. p vj. and 37. p xj.

The proportion may also here in part bee expressed by numbers: And yet a continuall is not required, as it was in the former.

In Plaines let the first example be, as followeth.

The cause of proportionall figures, for that twice two figures have the same reason doubled.

In Solids let this bee the second example. And yet here the figures are not proportionall unto the right lines, as before figures of equall heighth were unto their bases, but they themselves are proportionall one to another. And yet are they not proportionall in the same kinde of proportion.

The cause also is here the same, that was before: To witt, because twice two figures have the same reason trebled.

27. Figures filling a place, are those which being any way set about the same point, doe leave no voide roome.

This was the definition of the ancient Geometers, as appeareth out of Simplicius, in his commentaries upon the 8. chapter of Aristotle's iij. booke of Heaven: which kinde of figures Aristotle in the same place deemeth to bee onely ordinate, and yet not all of that kind. But only three among the Plaines, to witt a Triangle, a Quadrate, and a Sexangle: amongst Solids, two; the Pyramis, and the Cube. But if the filling of a place bee judged by right angles, 4. in a Plaine, and 8. in a Solid, the Oblong of plaines, and the

Octahedrum of Solids shall (as shall appeare in their places) fill a place; And yet is not this Geometrie of Aristotle accurate enough. But right angles doe determine this sentence, and so doth Euclide out of the angles demonstrate, That there are onely five ordinate solids; And so doth Potamon the Geometer, as Simplicus testifieth, demonstrate the question of figures filling a place. Lastly, if figures, by laying of their corners together, doe make in a Plaine 4. right angles, or in a Solid 8. they doe fill a place.

Of this probleme the ancient geometers have written, as we heard even now: And of the latter writers, Regiomontanus is said to have written accurately; And of this argument Maucolycus hath promised a treatise, neither of which as yet it hath beene our good hap to see.

Neither of these are figures of this nature, as in their due places shall be proved and demonstrated.

28. A round figure is that, all whose raies are equall.

Such in plaines shall the Circle be, in Solids the Globe or Spheare. Now this figure, the Round, I meane, of all Isoperimeters is the greatest, as appeared before at the [15. e]. For which cause Plato, in his Timæus or his Dialogue of the World said; That this figure is of all other the greatest. And therefore God, saith he, did make the world of a

sphearicall forme, that within his compasse it might the better containe all things: And Aristotle, in his Mechanicall problems, saith; That this figure is the beginning, principle, and cause of all miracles. But those miracles shall in their time God willing, be manifested and showne.

Rotundum, a Roundle, let it be here used for Rotunda figura, a round figure. And in deede Thomas Finkius or Finche, as we would call him, a learned Dane, sequestring this argument from the rest of the body of Geometry, hath intituled that his worke De Geometria rotundi, Of the Geometry of the Round or roundle.

29. The diameters of a roundle are cut in two by equall raies.

The reason is, because the halfes of the diameters, are the raies. Or because the diameter is nothing else but a doubled ray: Therefore if thou shalt cut off from the diameter so much, as is the radius or ray, it followeth that so much shall still remaine, as thou hast cutte of, to witt one ray, which is the other halfe of the diameter. Sn.

And here observe, That Bisecare, doth here, and in other places following, signifie to cutte a thing into two equall parts or portions; And so Bisegmentum, to be one such portion; And Bisectio, such a like cutting or division.

30. Rounds of equall diameters are equall. Out of the 1. d. iij.

Circles and Spheares are equall, which have equall diameters. For the raies, which doe measure the space betweene the Center and Perimeter, are equall, of which, being doubled, the Diameter doth consist. Sn.

The fifth Booke, of Ramus his Geometry, which is of Lines and Angles in a plaine Surface.

1. A lineate is either a Surface or a Body.

Lineatum, (or Lineamentum) a magnitude made of lines, as was defined at [1. e. iij]. is here divided into two kindes: which is easily conceived out of the said definition there, in which a line is excluded, and a Surface & a body are comprehended. And from hence arose the division of the arte Metriall into Geometry, of a surface, and Stereometry, of a body, after which maner Plato in his vij. booke of his Common-wealth, and Aristotle in the 7. chapter of the first booke of his Posteriorums, doe distinguish betweene Geometry and Stereometry: And yet the name of Geometry is used to signifie the whole arte of measuring in generall.

2. A Surface is a lineate only broade. 5. d j.

As here aeio. and uysr. The definition of a Surface doth comprehend the distance or dimension of a line, to

witt Length: But it addeth another distance, that is Breadth. Therefore a Surface is defined by some, as Proclus saith, to be a magnitude of two dimensions. But two doe not so specially and so properly define it. Therefore a Surface is better defined, to bee a magnitude onely long and broad. Such, saith Apollonius, are the shadowes upon the earth, which doe farre and wide cover the ground and champion fields, and doe not enter into the earth, nor have any manner of thicknesse at all.

Epiphania, the Greeke word, which importeth onely the outter appearance of a thing, is here more significant, because of a Magnitude there is nothing visible or to bee seene, but the surface.

3. The bound of a surface is a line. 6. d j.

The matter in Plaines is manifest. For a three cornered surface is bounded with 3. lines: A foure cornered surface, with foure lines, and so forth: A Circle is bounded with one line. But in a Sphearicall surface the matter is not so plaine: For it being whole, seemeth not to be bounded with a line. Yet if the manner of making of a Sphearicall surface, by the conversiō or turning about of a semiperiphery, the beginning of it, as also the end, shalbe a line, to wit a semiperiphery: And as a point doth not only actu, or indeede bound and end a line: But is potentia, or in power, the middest of it: So also a line boundeth a Surface actu, and an innumerable company of lines may be taken or supposed to be throughout the whole surface. A Surface therefore is made by the motion of a line, as a Line was made by the motion of a point.

4. A Surface is either Plaine or Bowed.

The difference of a Surface, doth answer to the difference of a Line, in straightnesse and obliquity or crookednesse.

Obliquum, oblique, there signified crooked; Not right or straight: Here, uneven or bowed, either upward or downeward. Sn.

5. A plaine surface is a surface, which lyeth equally betweene his bounds, out of the 7. d j.

As here thou seest in aeio. That therefore a Right line doth looke two contrary waies, a Plaine surface doth looke all about every way, that a plaine surface should, of all surfaces within the same bounds, be the shortest: And that the middest thereof should hinder the sight of the extreames. Lastly, it is equall to the dimension betweene the lines: It may also by one right line every way applyed be tryed, as Proclus at this place doth intimate.

Planum, a Plaine, is taken and used for a plaine surface: as before Rotundum, a Round, was used for a round figure.

Therefore,

6. From a point unto a point we may, in a plaine surface, draw a right line, 1 and 2. post. j.

Three things are from the former ground begg'd: The first is of a Right line. A right line and a periphery were in the ij. booke defined: But the fabricke or making of them both, is here said to bee properly in a plaine.

The fabricke or construction of a right line is the 1. petition. And justly is it required that it may bee done onely upon a plaine: For in any other surface it were in vaine to aske it. For neither may wee possibly in a sphericall betweene two points draw a right line: Neither may wee possibly in a Conicall and Cylindraceall betweene any two points assigned draw a right line. For from the toppe

unto the base that in these is only possible: And then is it the bounde of the plaine which cutteth the Cone and Cylinder. Therefore, as I said, of a right plaine it may onely justly bee demanded: That from any point assigned, unto any point assigned, a right line may be drawne, as here from a unto e.

Now the Geometricall instrument for the drawing of a right plaine is called Amussis, & by Petolemey, in the 2. chapter of his first booke of his Musicke, Regula, a Rular, such as heere thou seest.

And from a point unto a point is this justly demanded to be done, not unto points; For neither doe all points fall in a right line: But many doe fall out to be in a crooked line. And in a Spheare, a Cone & Cylinder, a Ruler may be applyed, but it must be a sphearicall, Conicall, or Cylindraceall. But by the example of a right line doth Vitellio, 2 p j. demaund that betweene two lines a surface may be extended: And so may it seeme in the Elements, of many figures both plaine and solids, by Euclide to be demanded; That a figure may be described, at the 7. and 8. e ij. Item that a figure may be made vp, at the 8. 14. 16. 23. 28. p. vj. which are of Plaines. Item at the 25. 31. 33. 34. 36. 49. p. xj. which are of Solids. Yet notwithstanding a plaine surface, and a plaine body doe measure their rectitude by a right line, so that jus postulandi, this right of begging to have a thing granted may seeme primarily to bee in a right plaine line.

Now the Continuation of a right line is nothing else, but the drawing out farther of a line now drawne, and that from a point unto a point, as we may continue the right line ae. unto i. wherefore the first and second Petitions of Euclide do agree in one.

And

7. To set at a point assigned a Right line equall to another right line given: And from a greater, to cut off a part equall to a lesser. 2. and 3. p j.

As let the Right line given be ae. And to i. a point assigned, grant that io. equall to the same ae. may bee set. Item, in the second example, let ae. bee greater then io. And let there be cut off from the same ae. by applying of a rular made equall to io. the lesser, portion au. as here. For if any man shall thinke that this ought only to be don in the minde, hee also, as it were, beares a ruler in his minde, that he may doe it by the helpe of the ruler. Neither is the fabricke in deede, or making of one right equal to another: And the cutting off from greater Right line, a portion equall to a lesser, any whit harder, then it was, having a point and a distance given, to describe a circle: Then having a Triangle, Parallelogramme, and semicircle given, to describe or make a Cone, Cylinder, and spheare, all which notwithstanding Euclide did account as principles.

Therefore,

8. One right line, or two cutting one another, are in the same plaine, out of the 1. and 2. p xj.

One Right line may bee the common section of two plaines: yet all or the whole in the same plaine is one: And all the whole is in the same other: And so the whole is the same plaine. Two Right lines cutting one another, may bee in two plaines cutting one of another; But then a plaine may be drawne by them: Therefore both

of them shall be in the same plaine. And this plaine is geometrically to be conceived: Because the same plaine is not alwaies made the ground whereupon one oblique line, or two cutting one another are drawne, when a periphery is in a sphearicall: Neither may all peripheries cutting one another be possibly in one plaine.

And

9. With a right line given to describe a peripherie.

This fabricke or construction is taken out of the 3. Petition which is thus. Having a center and a distance given to describe, make, or draw a circle. But here the terme or end of a circle is onely sought, which is better drawne out of the definition of a periphery, at the [10. e ij]. And in a plaine onely may that conversion or turning about of a right line bee made: Not in a sphearicall, not in a Conicall, not in a Cylindraceall, except it be in top, where notwithstanding a periphery may bee described. Therefore before (to witt at the said [10. e ij].) was taught the generall fabricke or making of a Periphery: Here we are informed how to discribe a Plaine periphery, as here.

Now as the Rular was the instrument invented and used for the drawing of a right line: so also may the same Rular, used after another manner, be the instrument to describe or draw a periphery withall. And indeed such is that instrument used by the Coopers (and other like artists) for the rounding of their bottomes of their tubs, heads of barrells and otherlike vessells: But the Compasses, whether straight shanked or bow-legg'd, such as here thou seest, it skilleth not, are for al purposes and practises, in this case the best and readiest. And in deed the Compasses, of all

geometricall instruments, are the most excellent, and by whose help famous Geometers have taught: That all the problems of geometry may bee wrought and performed: And there is a booke extant, set out by John Baptist, an Italian, teaching, How by one opening of the Compasses all the problems of Euclide may be resolved: And Jeronymus Cardanus, a famous Mathematician, in the 15. booke of his Subtilties, writeth, that there was by the helpe of the Compasses a demonstration of all things demonstrated by Euclide, found out by him and one Ferrarius.

Talus, the nephew of Dædalus by his sister, is said in the viij. booke of Ovids Metamorphosis, to have beene the inventour of this instrument: For there he thus writeth of him and this matter:—Et ex uno duo ferrea brachia nodo: Iunxit, ut æquali spatio distantibus ipsis: Altera pars staret, pars altera duceret orbem.

Therfore

10. The raies of the same, or of an equall periphery, are equall.

The reason is, because the same right line is every where converted or turned about. But here by the Ray of the periphery, must bee understood the Ray the figure contained within the periphery.

11. If two equall peripheries, from the ends of equall shankes of an assigned rectilineall angle, doe meete before it, a right line drawne from the meeting of them unto the toppe or point of the angle, shall cut it into two equall parts. 9. p j.

Hitherto we have spoken of plaine lines: Their affection followeth, and first in the Bisection or dividing of an Angle into two equall parts.

Let the right lined Angle to bee divided into two equall parts bee eai. whose equall shankes let them be ae. and ai. (or if they be unequall, let them be made equall, by the [7 e].) Then two equall peripheries from the ends e and i. meet before the Angle in o. Lastly, draw a line from o. unto a. I say the angle given is divided into two equall parts. For by drawing the right lines oe. and oi. the angles oae. and oai. equicrurall, by the grant, and by their common side ao. are equall in base eo. and io. by the [10 e] (Because they are the raies of equall peripheries.) Therefore by the [7. e iij]. the angles oae. and oai. are equall: And therefore the Angle eai. is equally divided into two parts.

12. If two equall peripheries from the ends of a right line given, doe meete on each side of the same, a right line drawne from those meetings, shall divide the right line given into two equall parts. 10. p j.

Let the right line given bee ae. And let two equall peripheries from the ends a. and e. meete in i. and o. Then from those meetings let the right line io. be drawne. I say, That ae. is divided into two equall parts, by the said line thus

drawne. For by drawing the raies of the equall peripheries ia. and ie. the said io. doth cut the angle aie. into two equall parts, by the [11. e]. Therefore the angles aiu. and uie. being equall and equicrurall (seeing the shankes are the raies of equall peripheries, by the grant.) have equall bases au. and ue. by the [7. e iij]. Wherefore seeing the parts au. and ue. are equall, ae. the assigned right line is divided into two equall portions.

13. If a right line doe stand perpendicular upon another right line, it maketh on each side right angles: And contrary wise.

A right line standeth upon a right line, which cutteth, and is not cut againe. And the Angles on each side, are they which the falling line maketh with that underneath it, as is manifest out of Proclus, at the 15. pj. of Euclide; As here ae. the line cut: and io. the insisting line, let them be perpendicular; The angles on each side, to witt aio. and eio. shall bee right angles, by the [13. e iij].

The Rular, for the making of straight lines on a plaine, was the first Geometricall instrument: The Compasses, for the describing of a Circle, was the second: The Norma or Square for the true erecting of a right line in the same plaine upon another right line, and then of a surface and body, upon a surface or body, is the third. The figure therefore is thus.

Now Perpendiculū, an instrument with a line & a plummet of leade appendant upon it, used of Architects, Carpenters, and Masons, is meerely physicall: because heavie things

naturally by their weight are in straight lines carried perpendicularly downeward. This instrument is of two sorts: The first, which they call a Plumbe-rule, is for the trying of an erect perpendicular, as whether a columne, pillar, or any other kinde of building bee right, that is plumbe unto the plaine of the horizont & doth not leane or reele any way. The second is for the trying or examining of a plaine or floore, whether it doe lye parallell to the horizont or not. Therefore when the line from the right angle, doth fall upon the middle of the base; it shall shew that the length is equally poysed. The Latines call it Libra, or Libella, a ballance: of the Italians Livello, and vel Archipendolo, Achildulo: of the French, Nivelle, or Niueau: of us a Levill.

Therefore

14. If a right line do stand upon a right line, it maketh the angles on each side equall to two right angles: and contrariwise out of the 13. and 14. p j.

For two such angles doe occupy or fill the same place that two right angles doe: Therefore

they are equall to them by the 11. e j. If the insisting line be perpendicular unto that underneath it, it then shall make 2. right angles, by the [13. e]. If it bee not perpendicular, & do make two oblique angles, as here aio. and oie. are yet shall they occupy the same place that two right angles doe: And therefore they are equall to two right angles, by the same.

The converse is forced by an argument ab impossibli, or ab absurdo, from the absurdity which otherwise would follow of it: For the part must otherwise needes bee equall to the whole. Let therefore the insisting or standing line which maketh two angles aeo. and aeu. on each side equall to two right angles, be ae. I say that oe. and ei. are but one right line. Otherwise let oe. bee continued unto u. by the [6. e]. Now by the [14. e.] or next former element, aeo. & aeu. are equall to two right angles; To which also oea. & aei. are equall by the grant: Let aeo. the common angle be taken away: then shall there be left aeu. equall to aei. the part to the whole, which is absurd and impossible. Herehence is it certaine that the two right lines oe, and ei, are in deede but one continuall right line.

And

15. If two right lines doe cut one another, they doe make the angles at the top equall and all equall to foure right angles. 15. p j.

Anguli ad verticem, Angles at the top or head, are called Verticall angles which have their toppes meeting in the same point. The Demonstration is: Because the lines cutting one another, are either perpendiculars, and then all

right angles are equall as heere: Or else they are oblique, and then also are the verticalls equall, as are aui, and oue: And againe, auo, and iue. Now aui, and oue, are equall, because by the [14. e.] with auo, the common angle, they are equall to two right angles: And therefore they are equall betweene themselves. Wherefore auo, the said common angle beeing taken away, they are equall one to another.

And

16. If two right lines cut with one right line, doe make the inner angles on the same side greater then two right angles, those on the other side against them shall be lesser then two right angles.

As here, if auy, and uyi, bee greater then two right angles euy, and uyo, shall bee lesser then two right angles.

17. If from a point assigned of an infinite right line given, two equall parts be on each side cut off: and then from the points of those sections two equall circles doe meete, a right line drawne from their meeting unto the point assigned, shall bee perpendicular unto the line given. 11. p j.

As let a, be the point assigned of the infinite line given: and from that on each side, by the [7. e.] cut off equall

portions ae, and ai, Then let two equall peripheries from the points e, and i, meete, as in o, I say that a right line drawne from o, the point of the meeting of the peripheries. unto a. the point given, shalbe perpendicular upon the line given. For drawing the right lines oe, & oi, the two angles eao, and iao, on each side, equicrurall by the construction of equall segments on each side, and oa, the common side, are equall in base by the [9. e]. And therefore the angles themselves shall be equall, by the [7. e iij]. and therefore againe, seeing that ao, doth lie equall betweene the parts ea, and ia, it is by the [13. e ij]. perpendicular upon it.

18. If a part of an infinite right line, bee by a periphery for a point given without, cut off a right line from the said point, cutting in two the said part, shall bee perpendicular upon the line given. 12. p j.

Of an infinite right line given, let the part cut off by a periphery of an externall center be ae: And then let io, cut the said part into two parts by the [12. e]. I say that io is perpendicular unto the said infinite right line. For it standeth upright, and maketh aoi, and eoi, equall angles, for the same cause, whereby the next former perpendicular was demonstrated.

19. If two right lines drawne at length in the same plaine doe never meete, they are parallells. è 35. d j.

Thus much of the Perpendicularity of plaine right lines: Parallelissmus, or their parallell equality doth follow.

Euclid did justly require these lines so drawne to be granted paralels: for then shall they be alwayes equally distant, as here ae. and io.

Therefore

20. If an infinite right line doe cut one of the infinite right parallell lines, it shall also cut the other.

As in the same example uy. cutting ae. it shall also cut io. Otherwise, if it should not cut it, it should be parallell unto it, by the [18 e]. And that against the grant.

21. If right lines cut with a right line be pararellells, they doe make the inner angles on the same side equall to two right angles: And also the alterne angles equall betweene themselves: And the outter, to the inner opposite to it: And contrariwise, 29, 28, 27. p 1.

The paralillesme, or parallell-equality of right lines cut with a right line, concludeth a threefold equality of angles: And the same is againe of each of them concluded. Therefore in this one element there are sixe things taught; all which are manifest if a perpendicular, doe fall

upon two parallell lines. The first sort of angles are in their owne words plainely enough expressed. But the word Alternum, alterne [or alternate, H.] here, as Proclus saith, signifieth situation, which in Arithmeticke signified proportion, when the antecedent was compared to the consequent; notwithstanding the metaphor answereth fitly. For as an acute angle is unto his successively following obtuse; So on the other part is the acute unto his successively following obtuse: Therefore alternly, As the acute unto the acute: so is the obtuse, unto the obtuse. But the outter and inner are opposite, of the which the one is without the parallels; the other is within on the same part not successively; but upon the same right line the third from the outer.

The cause of this threefold propriety is from the perpendicular or plumb-line, which falling upon the parallells breedeth and discovereth all this variety: As here they are right angles which are the inner on the same part or side: Item, the alterne angles: Item the inner and the outter: And therefore they are equall, both, I meane, the two inner to two right angles: and the alterne angles between themselvs: And the outter to the inner opposite to it.

If so be that the cutting line be oblique, that is, fall not upon them plumbe or perpendicularly, the same shall on the contrary befall the parallels. For by that same obliquation or slanting, the right lines remaining and the angles unaltered, in like manner both one of the inner, to wit, euy, is made obtuse, the other, to wit, uyo, is made acute: And the alterne angles are made acute and obtuse: As also the outter and inner opposite are likewise made acute and obtuse.

If any man shall notwithstanding say, That the inner angles are unequall to two right angles: By the same argument may he say (saith Ptolome in Proclus) That on each side they be both greater than two right angles, and also lesser: As in the parallel right lines ae and io, cut with

the right line uy, if thou shalt say that auy and iyu, are greater then two right angles, the angles on the other side, by the [16 e], shall be lesser then two right angles, which selfesame notwithstanding are also, by the gainesayers graunt, greater then two right angles, which is impossible.

The same impossibility shall be concluded, if they shall be sayd, to be lesser than two right angles.

The second and third parts may be concluded out of the first. The second is thus: Twise two angles are equall to two right angles oyu, and euy, by the former part: Item, auy, and euy, by the [14 e]. Therefore they are equall betweene themselves. Now from the equall, Take away euy, the common angle, And the remainders, the alterne angles, at u, and y shall be least equall.

The third is thus: The angles euy, and oys, are equall to the same uyi, by the second propriety, and by the [15 e]. Therefore they are equall betweene themselves.

The converse of the first is here also the more manifest by that light of the common perpendicular, And if any man shall thinke, That although the two inner angles be equall to two right angles, yet the right may meete, as if those equall angles were right angles, as here; it must needes be that two right lines divided by a common perpendicular, should both leane, the one this way, the other that way, or at least one of them, contrary to the [13 e ij].

If they be oblique angles, as here, the lines one slanting or

obliquely crossing one another, the angles on one side will grow lesse, on the other side greater. Therefore they would not be equall to two right angles, against the graunt.

From hence the second and third parts may be concluded. The second is thus: The alterne angles at u and y, are equall to the foresayd inner angles, by the 14 e: Because both of them are equall to the two right angles: And so by the first part the second is concluded.

The third is therefore by the second demonstrated, because the outter oys, is equall to the verticall or opposite angle at the top, by the [15 e]. Therefore seeing the outter and inner opposite are equall, the alterne also are equall.

Wherefore as Parallelismus, parallell-equality argueth a three-fold equality of angles: So the threefold equality of angles doth argue the same parallel-equality.

Therefore,

22. If right lines knit together with a right line, doe make the inner angles on the same side lesser than two right Angles, they being on that side drawne out at length, will meete.

As here ae, and io, knit together with eo, doe make two angles aeo, and ioe, lesser than two right angles: They shall therefore, I say, meete if they be continued out that wayward. The assumption and complexion is out of the [21 e], of right lines in the same plaine. If right lines cut with a right line be parallels, they doe make the inner angles on the same part equall to two right-angles. Therefore if they doe not make them equall, but lesser, they shall not be parallel, but shall meete.

And

23. A right line knitting together parallell right lines, is in the same plaine with them. 7 p xj.

As here uy, knitting or joyning together the two parallels ae, and io, is in the same plaine with them as is manifest by the [8 e].

And

24. If a right line from a point given doe with a right line given make an angle, the other shanke of the angle equalled and alterne to the angle made, shall be parallell unto the assigned right line. 31 p j.

As let the assigned right line be ae: And the point given, let it be i. From which the right line, making with the assigned ae, the angle, ioe, let it be io: To the which at i, let the alterne angle oiu, be made equall: The right line ui, which is the other shanke, is parallel to the assigned ae.

An angle, I confesse, may bee made equall by the first propriety: And so indeed commonly the Architects and Carpenters doe make it, by erecting of a perpendicular. It may also againe in like manner be made by the outter angle: Any man may at his pleasure use which hee shall thinke good: But that here taught we take to be the best.

And

25. The angles of shanks alternly parallell, are equall. Or Thus, The angles whose alternate feete are parallells, are equall. H.

This consectary is drawne out of the third property of

the [21 e]. The thing manifest in the example following, by drawing out, or continuing the other shanke of the inner angle. But Lazarus Schonerus it seemeth doth thinke the adverbe alterne, (alternely or alternately) to be more then needeth: And therefore he delivereth it thus: The angles of parallel shankes are equall.

And

26. If parallels doe bound parallels, the opposite lines are equall è 34 p. j. Or thus: If parallels doe inclose parallels, the opposite parallels are equall. H.

Otherwise they should not be parallell. This is understood by the perpendiculars, knitting them together, which by the definition are equall betweene two parallells: And if of perpendiculars they bee made oblique, they shall notwithstanding remaine equall, onely the corners will be changed.

And

27. If right lines doe joyntly bound on the same side equall and parallell lines, they are also equall and parallell.

This element might have beene concluded out of the next precedent: But it may also be learned out of those

which went before. As let ae, and io, equall parallels be bounded joyntly of ai, and eo: and let ei be drawn. Here because the right line ei falleth upon the parallels ae, and io, the alterne angles aei and eio, are equall, by the [21 e]. And they are equall in shankes ae, and io, by the grants, and ei, is the common shanke: Therefore they are also equall in base ai, and eo, by the [7 e iij]. This is the first: Then by [21 e], the alterne angles eia, and ieo, are equall betweene themselves: And those are made by ai and eo, cut by the right line ei: Therefore they are parallell; which was the second.

On the same part or side it is sayd, least any man might understand right lines knit together by opposite bounds as here.

28. If right lines be cut joyntly by many parallell right lines, the segments betweene those lines shall bee proportionall one to another, out of the 2 p vj and 17 p xj.

Thus much of the Perpendicle, and parallell equality of plaine right lines: Their Proportion is the last thing to be considered of them.

The truth of this element dependeth upon the nature of the parallells: And that throughout all kindes of equality and inequality, both greater and lesser. For if the lines thus cut be perpendiculars, the portions

intercepted betweene the two parallels shall be equall: for common perpendiculars doe make parallell equality, as before hath beene taught, and here thou seest.

If the lines cut be not parallels, but doe leane one toward another, the portions cut or intercepted betweene them will not be equall, yet shall they be proportionall one to another. And looke how much greater the line thus cut is: so much greater shall the intersegments or portions intercepted be. And contrariwise, Looke how much lesse: so much lesser shall they be.

The third parallell in the toppe is not expressed, yet must it be understood.

This element is very fruitfull: For from hence doe arise and issue, First the manner of cutting a line according to any rate or proportion assigned: And then the invention or way to finde out both the third and fourth proportionalls.

29. If a right line making an angle with another right line, be cut according to any reason [or proportion] assigned, parallels drawne from the ends of the segments, unto the end of the sayd right line given and unto some contingent point in the same, shall cut the line given according to the reason given.

Schoner hath altered this Consectary, and delivereth it

thus: If a right line making an angle with a right line given, and knit unto it with a base, be cut according to any rate assigned, a parallell to the base from the ends of the segments, shall cut the line given according to the rate assigned. 9 and 10 p vj.

Punctum contingens, A contingent point, that is falling or lighting in some place at al adventurs, not given or assigned.

This is a marvelous generall consectary, serving indifferently for any manner of section of a right line, whether it be to be cut into two parts, or three parts, or into as many parts, as you shall thinke good, or generally after what manner of way soever thou shalt command or desire a line to be cut or divided.

Let the assigned Right line to be cut into two equall parts be ae. And the right line making an angle with it, let it be the infinite right line ai. Let ao, one portion thereof be cut off. And then by the [7 e], let oi, another part thereof be taken equall to it. And lastly, by the [24 e], draw parallels from the points i, and o, unto e, the end of the line given, and to u; a contingent point therein. Now the third parallell is understood by the point a, neither is it necessary that it should be expressed. Therefore the line ae, by the [28], is cut into two equall portions: And as ao, is to oi: So is au, to ue. But ao, and oi, are halfe parts. Therefore au, and ue, are also halfe parts.

And here also is the [12 e] comprehended, although not in the same kinde of argument, yet in effect the same. But that argument was indeed shorter, although this be more generall.

Now let ae be cut into three parts, of which the first let it bee

the halfe of the second: And the second, the halfe of the third: And the conterminall or right line making an angle with the sayd assigned line, let it be cut one part ao: Then double this in ou: Lastly let ui be taken double to ou, and let the whole diagramme be made up with three parallels ie, uy, and os, The fourth parallell in the toppe, as afore-sayd, shall be understood. Therefore that section which was made in the conterminall line, by the [28 e], shall be in the assigned line: Because the segments or portions intercepted are betweene the parallels.

And

30. If two right lines given, making an angle, be continued, the first equally to the second, the second infinitly, parallels drawne from the ends of the first continuation, unto the beginning of the second, and some contingent point in the same, shall intercept betweene them the third proportionall. 11. p vj.

Let the right lines given, making an angle, be ae, and ai: and ae, the first, let it be continued equally to the same ai, and the same ai, let it be drawne out infinitly: Then the parallels ei, and ou, drawne from the ends of the first continuation, unto i, the beginning of the second: and u, a contingent point in the second, doe cut off iu, the third proportionall sought. For by the [28 e], as ae, is unto eo, so is ai, unto iu.

And

31. If of three right lines given, the first and the third making an angle be continued, the first equally to the second, and the third infinitly; parallels drawne

from the ends of the first continuation, unto the beginning of the second, and some contingent point, the same shall intercept betweene them the fourth proportionall. 12. p vj.

Let the lines given be these: The first ae, the second ei, the third ao, and let the whole diagramme be made up according to the prescript of the consectary. Here by [28. e], as ae, is to ei so is ao, to ou. Thus farre Ramus.

Lazarus Schonerus, who, about some 25. yeares since, did revise and augment this worke of our Authour, hath not onely altered the forme of these two next precedent consectaries: but he hath also changed their order, and that which is here the second, is in his edition the third: and the third here, is in him the second. And to the former declaration of them, hee addeth these words: From hence, having three lines given, is the invention of the fourth proportionall; and out of that, having two lines given, ariseth the invention of the third proportionall.

2 Having three right lines given, if the first and the third making an angle, and knit together with a base, be continued, the first equally to the second; the third infinitly; a parallel from the end of the second, unto the continuation of the third, shall intercept the fourth proportionall. 12. p vj.

The Diagramme, and demonstration is the same with our [31. e] or 3 c of Ramus.

3 If two right lines given making an angle, and knit together with a base, be continued, the first equally to the second, the second infinitly; a parallell to the base from the end of the first continuation unto the second, shall intercept the third proportionall. 11. p vj.

The Diagramme here also, and demonstration is in all

respects the same with our [30 e], or 2 c of Ramus.

Thus farre Ramus: And here by the judgement of the learned Finkius, two elements of Ptolomey are to be adjoyned.

32 If two right lines cutting one another, be againe cut with many parallels, the parallels are proportionall unto their next segments.

It is a consectary out of the [28 e]. For let the right lines ae. and ai, cut one another at a, and let two parallell lines uo, and ei, cut them; I say, as au, is to uo, so ae, is to ei. For from the end i, let is, be erected parallell to ae, and let uo, be drawne out untill it doe meete with it. Then from the end s, let sy, be made parallell to ai: and lastly, let ea, be drawne out, untill it doe meete with it. Here now ay, shall be equall to the right line is, that is, by the [26. e], to ue: and at length, by the [28. e], as ua, is to uo; so is ay, that is, ue, to os. Therefore, by composition or addition of proportions, as ua, is unto uo, so ua, and ue, shall be unto uo, and os, that is, ei, by the [27. e].

The same demonstration shall serve, if the lines do crosse one another, or doe vertically cut one another, as in the same diagramme appeareth. For if the assigned ai, and us, doe cut one another vertically in o, let them be cut with the parallels au, and si: the precedent fabricke or figure being made up, it shall be by [28. e.] as au, is unto ao, the segment next unto it: so ay, that is, is, shall be unto oi, his next segment.

The [28. e] teacheth how to finde out the third and fourth proportionall: This affordeth us a meanes how to find out

the continually meane proportionall single or double.

Therefore

33. If two right lines given be continued into one, a perpendicular from the point of continuation unto the angle of the squire, including the continued line with the continuation, is the meane proportionall betweene the two right lines given.

A squire (Norma, Gnomon, or Canon) is an instrument consisting of two shankes, including a right angle. Of this we heard before at the [13. e]. By the meanes of this a meane proportionall unto two lines given is easily found: whereupon it may also be called a Mesolabium, or Mesographus simplex, or single meane finder.

Let the two right lines given, be ae, and ei. The meane proportional between these two is desired. For the finding of which, let it be granted that as ae, is to eo, so eo, is to ei: therefore let ae, be continued or drawne out unto i, so that ei, be equall to the other given. Then from e, the point of the continuation, let eo, an infinite perpendicular be erected. Now about this perpendicular, up and downe, this way and that way, let the squire ao, be moved, so that with his angle it may comprehend at eo, and with his shanks it may include the whole right line ai. I say that eo, the segment of the perpendicular, is the meane proportionall between

ae, and ei, the two lines given. For let ea, be continued or drawne out into u, so that the continuation au, be equall unto eo: and unto a, the point of the continuation, let the angle uas, be made equall, and equicrurall to the angle oei, that is, let the shanke as, be made equall to the shanke ei. Wherefore knitting u, and s, together, the right lines us, and oi, shall be equall; and the angles eoi, aus, by the [7. e iij]. And by the [21. e], the lines sa, and oe, are parallell: and the angle sao, is equall to the angle aoe. But the angles sae, and aoi, are right angles by the Fabricke and by the grant; and therefore they are equall, by the [14. e iij]. Wherefore the other angles oae, and eoi, that is, sua, are equall. And therefore by the [21. e.] us, and ao are parallell; and us, and eo, continued shall meete, as here in y: and by the [26. e.] oy, and as are equall. Now, by the [32. e.] as ue, is to ua, so is ey, to as. Therefore by subduction or subtraction of proportions, as ea, is to ua, so is eo, that is, ua, to oy, that is as.

And

34 If two assigned right lines joyned together by their ends rightanglewise, be continued vertically; a square falling with one of his shankes, and another to it parallell and moveable upon the ends of the assigned, with the angles upon the continued lines, shall cut betweene them from the continued two meanes continually proportionall to the assigned.

The former consectary was of a single mesolabium; this is of a double, whose use in making of solids, to this or that bignesse desired is notable.

Let the two lines assigned be ae, and ei; and let there be two meane right lines, continually proportionall betweene them sought, to wit, that may be as ae, is unto

one of the lines found; so the same may be unto the second line found. And as that is unto this, so this may be unto ei. Let therefore ae, and ei, be joyned rightanglewise by their ends at e; and let them be infinite continued, but vertically, that is, from that their meeting from the lines ward, from ei, towards u, but ae, towards o. Now for the rest, the construction; it was Plato's Mesographus; to wit, a squire with the opposits parallell. One of his sides au, moueable, or to be done up and downe, by an hollow riglet in the side adjoyning. Therefore thou shalt make thee a Mesographus, if unto the squire thou doe adde one moveable side, but so that how so ever it be moved, it be still parallell unto the opposite side [which is nothing else, but as it were a double squire, if this squire be applied unto it; and indeed what is done by this instrument, may also be done by two squires, as hereafter shall be shewed.] And so long and oft must the moveable side be moved up and downe, untill with the opposite side it containe or touch the ends of the assigned, but the angles must fall precisely upon the continued lines: The right lines from the point of the continuation, unto the corners of the squire, are the two meane proportionals sought.

As if of the Mesographus auoi, the moveable side be au;

thus thou shalt move up and downe, untill the angles u, and o, doe hit just upon the infinite lines; and joyntly at the same instant ua, and oi, may touch the ends of the assigned a, and i. By the former consectary it shall be as ei, is to eo, so eo, shall be unto eu: and as eo, is to eu, so shall eu, be unto ea.

And thus wee have the composition and use, both of the single and double Mesolabium.

35. If of foure right lines, two doe make an angle, the other reflected or turned backe upon themselves, from the ends of these, doe cut the former; the reason of the one unto his owne segment, or of the segments betweene themselves, is made of the reason of the so joyntly bounded, that the first of the makers be joyntly bounded with the beginning of the antecedent made; the second of this consequent joyntly bounded with the end; doe end in the end of the consequent made.

Ptolomey hath two speciall examples of this Theorem: to those Theon addeth other foure.

Let therefore the two right lines be ae, and ai: and from the ends of these other two reflected, be iu, and eo, cutting themselves in y; and the two former in u, and o. The reason of the particular right lines made shall be as

the draught following doth manifest. In which the antecedents of the makers are in the upper place: the consequents are set under neathe their owne antecedents.

The businesse is the same in the two other, whether you doe crosse the bounds or invert them.

Here for demonstrations sake we crave no more, but that from the beginning of an antecedent made a parallell be drawne to the second consequent of the makers, unto one of the assigned infinitely continued: then the multiplied proportions shall be,

The Antecedent, the Consequent; the Antecedent, the

Consequent of the second of the makers; every way the reason or rate is of Equallity.

The Antecedent; the Consequent of the first of the makers; the Parallell; the Antecedent of the second of the makers, by the [32. e]. Therefore by multiplication of proportions, the reason of the Parallell, unto the Consequent of the second of the makers, that is, by the fabricke or construction, and the [32. e.] the reason of the Antecedent of the Product, unto the Consequent, is made of the reason, &c. after the manner above written.

For examples sake, let the first speciall example be demonstrated. I say therefore, that the reason of ia, unto ao, is made of the reason of iu, unto uy, multiplied by the reason of ye, unto eo. For from the beginning of the Antecedent of the product, to wit, from the point i, let a line be drawne parallell to the right line ey, which shall meete with ae, continued or drawne out infinitely in n. Therefore, by the [32. e], as ia, is to ao: so is the parallell drawne to eo, the Consequent of the second of the makers. Therefore now the multiplied proportions are thus iu, uy, in, ey, by the 32. e: ye, eo, ey, eo. Therefore as the product of iu, by ye, is unto the product of uy, by eo: So in, is to eo, that is, ia, to ao.

So let the second of Ptolemy to be taught, which in our

Table aforegoing is the fifth. I say therefore that the reason of io, unto oa; is made of the reason of iy, unto yu, and the reason of ue, unto ea. For now againe, from the beginning of the Antecedent of the Product i, let a line be drawne parallell unto ea, the Consequent of the second of the Makers, which shall meete with eo, drawne out at length, in n: therefore, by the [32. e.] as io, is to ao; so is en, unto ea. Therefore now again the multiplied proportions are thus:

by the [32. e]. Therefore, by multiplication of proportions, the reason of en, unto ea, that is, of io, unto oa, is made of the reason of iy, unto yu, by the reason of ue, unto ea.

It shall not be amisse to teach the same in the examples of Theon. Let us take therefore the reason of the Reflex, unto the Segment; And of the segments betweene themselves; to wit, the 4. and 6. examples of our foresaid draught: I say therefore, that the reason of oe, unto ey, is made of the reason oa, unto ai, by the reason of iu, unto uy. For from the end o, to wit, from the beginning of the Antecedent of the product, let the right line no, be drawne parallell to uy. It shall be by the [32. e.] as oe, is to ey: so the parallell no, shall be to uy: but the reason of no, unto uy, is made of the reason of oa, unto ai, and of iu, unto uy: for the multiplied proportions are,

by the [32. e.]

Againe, I say, that the reason of ey, unto yo, is compounded of the reason of eu, unto ua, and of ai, unto io.

Theon here draweth a parallell from o, unto ui. By the generall fabricke it may be drawne out of e, unto ui.

It shall be therefore as ey, is unto yo, so en, shall be unto oi. Now the proportions multiplied are,

by the [32. e.]

Therefore the reason of en, unto io , that is of ey, unto

yo, shall be made of the foresaid reasons.

Of the segments of divers right lines, the Arabians have much under the name of The rule of sixe quantities. And the Theoremes of Althindus, concerning this matter, are in many mens hands. And Regiomontanus in his Algorithmus: and Maurolycus upon the 1 p iij. of Menelaus, doe make mention of them; but they containe nothing, which may not, by any man skillfull in Arithmeticke, be performed by the multiplication of proportions. For all those wayes of theirs are no more but speciall examples of that kinde of multiplication.

Of Geometry, the sixt Booke, of a Triangle.

1. Like plaines have a double reason of their homologall sides, and one proportionall meane, out of 20 p vj. and xj. and 18. p viij.

Or thus; Like plaines have the proportion of their correspondent proportionall sides doubled, & one meane proportionall: Hitherto wee have spoken of plaine lines and their affections: Plaine figures and their kindes doe follow in the next place. And first, there is premised a common corollary drawne out of the [24. e. iiij]. because in plaines there are but two dimensions.

2. A plaine surface is either rectilineall or obliquelineall, [or rightlined, or crookedlined. H.]

Straightnesse, and crookednesse, was the difference of lines at the [4. e. ij]. From thence is it here repeated and attributed to a surface, which is geometrically made of lines. That made of right lines, is rectileniall: that which is made of crooked lines, is Obliquilineall.

3. A rectilineall surface, is that which is comprehended of right lines.

A plaine rightlined surface is that which is on all sides inclosed and comprehended with right lines. And yet they are not alwayes right betweene themselves, but such lines as doe lie equally betweene their owne bounds, and without comparison are all and every one of them right lines.

4. A rightilineall doth make all his angles equall to right angles; the inner ones generally to paires from two forward: the outter always to foure.

Or thus: A right lined plaine maketh his angles equall unto right angles: Namely the inward angles generally, are equall unto the even numbers from two forward, but the outward angles are equall but to 4. right angles. H.

The first kinde I meane of rectilineals, that is a triangle doth make all his inner angles equall to two right angles, that is, to a binary, the first even number of right angles: the second, that is a quadrangle, to the second even number, that is, to a quaternary or foure: The third, that is, a Pentangle, of quinqueangle to the third, that is a senary of right angles, or 6. and so farre forth as thou seest in this Arithmeticall progression of even numbers,

Notwithstanding the outter angles, every side continued and drawne out, are alwayes equall to a quaternary of right angles, that is to foure. The former part being granted (for that is not yet demonstrated) the latter is from thence concluded: For of the inner angles, that of the outter, is easily proved. For the three angles of a triangle are equall to two right angles. The foure of a quadrangle to foure: of a quinquangle, to sixe: of a sexe angle, to eight: Of septangle, to tenne, and so forth, form a binarie by even numbers: Whereupon, by the [14. e. V]. a perpetuall quaternary of the outer angles is concluded.

5. A rectilineall is either a Triangle or a Triangulate.

As before of a line was made a lineate: so here in like manner of a triangle is made a triangulate.

6. A triangle is a rectilineall figure comprehended of three rightlines. 21. d j.

As here aei. A triangular figure is of Euclide defined from the three sides; whereupon also it might be called Trilaterum, that is three sided, of the cause: rather than Trianglum, three cornered, of the effect; especially seeing that three angles, and three sides

are not reciprocall or to be converted. For a triangle may have foure sides, as is Acidoides, or Cuspidatum, the barbed forme, which Zonodorus called Cœlogonion, or Cavangulum, an hollow cornered figure. It may also have both five, and sixe sides, as here thou seest. The name therefore of Trilaterum would more fully and fitly expresse the thing named: But use hath received and entertained the name of a triangle for a trilater: And therefore let it be still retained, but in that same sense:

7. A triangle is the prime figure of rectilineals.

A triangle or threesided figure is the prime or most simple figure of all rectilineals. For amongst rectilineall figures there is none of two sides: For two right lines cannot inclose a figure. What is meant by a prime figure, was taught at the [11. e. iiij].

And

8. If an infinite right line doe cut the angle of a triangle, it doth also cut the base of the same: Vitell. 29. t j.

9. Any two sides of a triangle are greater than the other.

Thus much of the difinition of a triangle; the reason or

rate in the sides and angles of a triangle doth follow. The reason of the sides is first.

Let the triangle be aei; I say, the side ai, is shorter, than the two sides ae, and ei, because by the [6. e ij], a right line is betweene the same bounds the shortest.

Therefore

10. If of three right lines given, any two of them be greater than the other, and peripheries described upon the ends of the one, at the distances of the other two, shall meete, the rayes from that meeting unto the said ends, shall make a triangle of the lines given.

Let it be desired that a triangle be made of these three lines, aei, given, any two of them being greater than the other: First let there be drawne an infinite right; From this let there be cut off continually three portions, to wit, ou, uy, and ys, equall to ae, and i, the three lines given. Then upon the ends y, and u, at the distances ou, and ys; let two peripheries meet in the point r. The rayes from that meeting unto the said ends, u, and y, shall make the triangle ury: for those rayes shall be equall to the right lines given, by the [10. e v].

And

11. If two equall peripheries, from the ends of a right line given, and at his distance, doe meete, lines

drawne from the meeting, unto the said ends, shall make an equilater triangle upon the line given. 1 p. j.

As here upon ae, there is made the equilater triangle, aei; And in like manner may be framed the construction of an equicrurall triangle, by a common ray, unequall unto the line given; and of a scalen or various triangle, by three diverse raies; all which are set out here in this one figure. But these specialls are contained in the generall probleme: neither doe they declare or manifest unto us any new point of Geometry.

12. If a right line in a triangle be parallell to the base, it doth cut the shankes proportionally: And contrariwise. 2 p vj.

Such therefore was the reason or rate of the sides in one triangle; the proportion of the sides followeth.

As here in the triangle aei, let ou, be parallell to the base; and let a third parallel be understood to be in the toppe a; therefore, by the [28. e. v]. the intersegments are proportionall.

The converse is forced out of

the antecedent: because otherwise the whole should be lesse than the part. For if ou, be not parallell to the base ei, then yu, is: Here by the grant, and by the antecedent, seeing ao, oe, ay, ye, are proportionall: and the first ao, is lesser than ay, the third: oe, the second must be lesser than ye, the fourth, that is the whole then the part.

13. The three angles of a triangle, are equall to two right angles. 32. p j.

Hitherto therefore is declared the comparison in the sides of a triangle. Now is declared the reason or rate in the angles, which joyntly taken are equall to two right angles.

The truth of this proposition, saith Proclus, according to common notions, appeareth by two perpendiculars erected upon the ends of the base: for looke how much by the leaning of the inclination, is taken from two right angles at the base, so much is assumed or taken in at the toppe, and so by that requitall the equality of two right angles is made; as in the triangle aei, let, by the [24. e v], ou, be parallell against ie. Here three particular angles, iao, iae, eau, are equall to two right lines; by the [14. e v]. But the inner angles are equall to the same three: For first, eai, is equall to it selfe: Then the other two are equall to their alterne angles, by the [24. e v].

Therefore

14. Any two angles of a triangle are lesse than two right angles.

For if three angles be equall to two right angles, then

are two lesser than two right angles.

And

15. The one side of any triangle being continued or drawne out, the outter angle shall be equal to the two inner opposite angles.

This is the rate of the inner angles in one and the same triangle: The rate of the outter with the inner opposite angles doth followe. As in the triangle aei, let the side ei, be continued or drawne out unto o; the two angles on each side aio and aie, are by the [14 e v]. equall to two right angles: and the three inner angles, are by the [13. e.] equall also to two right angles; take away aie, the common angle, and the outter angle aio, shall be left equall to the other two inner and opposite angles.

Therefore

16. The said outter angle is greater than either of the inner opposite angles. 16. p j.

This is a consectary following necessarily upon the next former consectary.

17. If a triangle be equicrurall, the angles at the base are equall: and contrariwise, 5. and 6. p. j.

The antecedent is apparent by the [7. e iij]. The converse is apparent by an impossibilitie, which otherwise must needs follow. For if any one shanke be greater than the other, as ae: Then by the [7. e v]. let oe, be cut off equall to it: and let oi, be drawne: then by [7. e iij]. the base oi, must

be equal to the base ae; but the base oi, is lesser than ae. For by the [9. e], ia; and ao, (to which ae, is equall, seeing that oe, is supposed to be equall to the same ai: and ao, is common to both) are greater than the said oi; therefore the same, oi, must be equall to the same ae, and lesser than the same, which is impossible. This was first found out by Thales Milesius.

Therefore

18. If the equall shankes of a triangle be continued or drawne out, the angles under the base shall be equall betweene themselves.

For the angles aei, and ieo: Item aie, and eiu, are equall to two right angles, by the [14. e v]. Therefore they are equall betweene themselves: wherefore if you shall take away the inner angles, equall betweene themselves, you shall leave the outter equall one to another.

And

19. If a triangle be an equilater, it is also an equiangle: And contrariwise.

It is a consectary out of the condition of an equicrurall triangle of two, both shankes and angles, as in the example aei, shall be demonstrated.

And

20. The angle of an equilater triangle doth

countervaile two third parts of a right angle. Regio. 23. p j.

For seeing that 3. angles are equall to 2. 1. must needs be equall to 2/3.

And

21. Sixe equilater triangles doe fill a place.

As here. For 2/3. of a right angle sixe lines added together doe make 12/3. that is foure right angles; but foure right angles doe fill a place by the [27. e. iiij].

22. The greatest side of a triangle subtendeth the greatest angle; and the greatest angle is subtended of the greatest side. 19. and 18. p j.

Subtendere, to draw or straine out something under another; and in this place it signifieth nothing else but to make a line or such like, the base of an angle, arch, or such like. And subtendi, is to become or made the base of an angle, arch, of a circle, or such like: As here, let ai, be a greater side than ae, I say the angle at e, shall be greater than that at i. For let there be cut off from ai, a portion equall to ae,; and let that be io: then the angle aei, equicrurall to the angle oie, shall be greater in base, by the grant. Therefore the angle shall be greater, by the [9 e iij].

The converse is manifest by the same figure: As let the angle aei, be greater than the angle aie. Therefore by the same, [9 e iij]. it is greater in base. For what is there spoken

of angles in generall, are here assumed specially of the angles in a triangle.

23. If a right line in a triangle, doe cut the angle in two equall parts, it shall cut the base according to the reason of the shankes; and contrariwise. 3. p vj.

The mingled proportion of the sides and angles doth now remaine to be handled in the last place.

Let the triangle be aei; and let the angle eai, be cut into two equall parts, by the right line ao: I say, as ea, is unto ai, so eo, is unto oi. For at the angle i, let the parallell iu, by the [24. e v]. be erected against ao; and continue or draw out ea, infinitly; and it shall by the [20. e v]. cut the same iu, in some place or other. Let it therefore cut it in u. Here, by the [28. e v]. as ea, is to au, so is eo, to oi. But au, is equall to ai, by the [17. e]. For the angle uia, is equall to the alterne angle oai, by the [21. e v]. And by the grant it is equall to oae, his equall: And by the [21. e v]. it is equall to the inner angle aui; and by that which is concluded it is equall to uia, his equall. Therefore by the [17. e], au, and ai, are equall. Therefore as ea, is unto ai, so is eo, unto oi.

The Converse likewise is demonstrated in the same figure. For as ea, is to ai; so is eo, to oi: And so is ea, to au, by the 12 e: therefore ai, and au, are equall, Item the angles eao, and oai, are equall to the angles at u, and i, by the [21. e v]. which are equall betweene themselves by the [17. e].

Of Geometry, the seventh Booke, Of the comparison of Triangles.

1. Equilater triangles are equiangles. 8. p. j.

Thus forre of the Geometry, or affections and reason of one triangle; the comparison of two triangles one with another doth follow. And first of their rate or reason, out of their sides and angles: Whereupon triangles betweene themselves are said to be equilaters and equiangles. First out of the equality of the sides, is drawne also the equalitie of the angles.

Triangles therefore are here jointly called equilaters, whose sides are severally equall, the first to the first, the second, to the second, the third to the third; although every severall triangle be inequilaterall. Therefore the equality of the sides doth argue the equality of the angles, by the [7. e iij]. As here.

2. If two triangles be equall in angles, either the two equicrurals, or two of equall either shanke, or base of two angles, they are equilaters, 4. and 26. p j.

Or thus; If two triangles be equall in their angles, either

in two angles contained under equall feet, or in two angles, whose side or base of both is equall, those angles are equilater. H.

This element hath three parts, or it doth conclude two triangles to be equilaters three wayes. 1. The first part is apparent thus: Let the two triangles be aei, and ouy; because the equall angles at a, and o, are equicrurall, therefore they are equall in base, by the [7. e iij].

2 The second thus: Let the said two triangles aei, and ouy, be equall in two angles a peece, at e, and i, and at u, and y. And let them be equall in the shanke ei, to uy. I say, they are equilaters. For if the side ae, (for examples sake) be greater than the side ou, let es, be cut off equall unto it; and draw the right line is. Here by the antecedent, the triangles sei, and ouy, shall be equiangles, and the angles sie, shall be equall to the angle oyu, to which

also the whole angle aie, is equall, by the grant. Therefore the whole and the part are equall, which is impossible. Wherefore the side ae, is not unequall but equall to the side ou: And by the antecedent or former part, the triangles aei, and ouy, being equicrurall, are equall, at the angle of the shanks: Therefore also they are equall in their bases ai, and oy.

3 The third part is thus forced: In the triangles aei, and ouy, let the angles at e, and i, and u, and y, be equall, as afore: And ae, the base of the angle at i, be equall to ou, the base of angle at y: I say that the two triangles given are equilaters. For if the side ei, be greater than the side uy, let es, be cut off equall to it, and draw the right line as. Therefore by the antecedent, the two triangles, aes, and ouy, equall in the angle of their equall shankes are equiangle: And the angle ase, is equall to the angle oyu, which is equall by the grant unto the angle aie. Therefore ase, is equall to aie, the outter to the inner, contrary to the [15. e. vj]. Therefore the base ei, is not unequall to the base uy, but equall. And therefore as above was said, the two triangles aei, and ouy, equall in the angle of their equall shankes, are equilaters.

3. Triangles are equall in their three angles.

The reason is, because the three angles in any triangle are

equall to two right angles, by the [13. e vj]. As here, the greatest triangle, all his corners joyntly taken, is equall to the least.

And yet notwithstanding it is not therefore to be thought to be equiangle to it: For Triangles are then equiangles, when the severall angles of the one, are equall to the severall angles of the other: Not when all joyntly are equall to all.

Therefore

4. If two angles of two triangles given be equall, the other also are equall.

All the three angles, are equall betweene themselves, by the [3 e]. Therefore if from equall you take away equall, those which shall remaine shall be equall.

5. If a right triangle equicrurall to a triangle be greater in base, it is greater in angle: And contrariwise. 25. and 24. p j.

Thus farre of the reason or rate of equality, in the sides and angles of triangles: The reason of inequality, taken out of the common and generall inequality of angles, doth

follow. The first is manifest, by the [9. e iij]. as here thou seest in aei and ouy.

6. If a triangle placed upon the same base, with another triangle, be lesser in the inner shankes, it is greater in the angle of the shankes.

This is a consectary drawne also out of the [10 e iij]. As here in the triangle aei, and aoi, within it and upon the same base. Or thus: If a triangle placed upon the same base with another triangle, be lesse then the other triangle, in regard of his feet, (those feete being conteined within the feete of the other triangle) in regard of the angle conteined under those feete, it is greater: H.

7. Triangles of equall heighth, are one to another as their bases are one to another.

Thus farre of the Reason or rate of triangles: The proportion of triangles doth follow; And first of a right line with the bases. It is a consectary out of the [16 e iiij].

Therefore

8. Upon an equall base, they are equall.

This was a generall consectary at the [16. e iiij]: From whence Archimedes concluded, If a triangle of equall heighth with many other triangles, have his base equall to the bases of them all, it is equall to them all: as here thou seest aei to be equall to the triangles aeo, uoy, syr, lrm, nmi. Here hence also thou mayst conclude, that Equilater triangles are equall: Because they are of equall heighth, and upon the same base.

And

9. If a right line drawne from the toppe of a triangle, doe cut the base into two equall parts, it doth also cut the triangle into two equall parts: and it is the diameter of the triangle.

As here thou seest: For the bisegments, or two equall portions thus cut are two triangles of equall heighth that is to say, they have one toppe common to both, within the same parallels) and upon equall bases: Therefore they are equall: And that right line shall be the diameter of the triangle, by the [5 e iiij], because it passeth by the center.

10. If a right line be drawne from the toppe of a triangle, unto a point given in the base (so it be not in the middest of it) and a parallell be drawne from the middest of the base unto the side, a right line drawne from the toppe of the sayd parallell unto the sayd point, shall cut the triangle into two equall parts.

Let the triangle given be aei: And let ao, cut the base ei, in o unequally: And let uy be parallell from u, the middest of the base, unto the sayd ao. I say that yo shall divide the triangle into two equall portions. For let au be knit together with a right line: That line, by the [9 e], shall divide the triangle into two equall parts. Now the two triangles ayu, and you, are equall by the 8 e; because they are of equall height, and upon the same base.

Take away ysu, the common triangle; And you shall leave asy, and osu, equall betweene themselves: The common right lined figure ysui, let it be added to both the sayd equall triangles: And then oyi, shall be equall to aui, the halfe part; And therefore aeoy, the other right lined figure, shall be the halfe of the triangle given.

11. If equiangled triangles be reciprocall in the shankes of the equall angle, they are equall: And contrariwise. 15. p. vj. Or thus, as the learned Mr. Brigges hath conceived it: If two triangles, having one angle, are reciprocall, &c.

Direct proportion in triangles, is such as hath in the former beene taught: Reciprocall proportion followeth. It is a consectary drawne out of the [18 e iiij]; which is manifest, as oft as the equall angle is a right angle: For then those shankes, [comprehending the equall angles,] are the heights and the bases; As here thou seest in the severed triangles. Notwithstanding in obliquangle triangles, although the shankes are not the heights, the cause of the truth hereof is the same. Yet if any man shall desire a demonstration of it, it is thus: Let therefore the diagramme or figure bee in the triangles aei, and aou: And the angles oau, and eai, let them be equall: And as ua is to ae, so let ia be unto ao: I say that the triangles aou, and eai, are equall. For eo being knit together with a right line, uao is unto oae, as ua is unto ae, by the 7 e:

And ia, unto ao, by the grant, is as eai is unto eao. Therefore uao, and eai, are unto eao proportionall: And therefore they are equall one to another.

The converse, is concluded by the same sorites, but by saying all backward. For ua unto ae is, as uao is unto oae, by the 7 e: And as eai, by the grant: Because they are equall: And as ia is unto ao, by the same, Wherefore ua is unto ae, as ia is unto ao.

12. If two triangles be equiangles, they are proportionall in shankes: And contrariwise: 4 and 5. p. vj.

The comparison both of the rate and proportion of triangles hath in the former beene taught: Their similitude remaineth for the last place. Which similitude of theirs consisteth indeed of the reason, or rate of their angles and proportion of the shankes. Therefore for just cause was the reason of the angles set first: Because from thence not onely their reason, but also their latter proportion is gathered. Let aei and iou, be two triangles equiangled: And let them be set upon the same line eiu, meeting or touching one another in the common point i. Then, seeing that the angles at e and i, are granted to bee equall, the lines oi, and ae, are parallel, by the [21 e v]. Therefore by the [22 e v] uo and ea, being continued, shall meete. Item, The right lines ai, and yu, by the [21 e v], are parallel, because the angle aie is equall to oui, the inner opposite to it. Therefore seeing that ai is parallell to the base yu, by the [21 e v], ea shall be to ay, that is, by the [26 e v], to io, as ei is to iu: And alternly, or crosse wayes, ea shall be to ei, as io is to iu. This is the first proportion. Item,

seeing that io is parallell to the base ye; yo, that is, by the [26 e v], ai shall bee unto ou, as ei, is unto iu: And crosse wise, as ai is unto ie, so is ou unto ui. This is the second proposition. Lastly, equiordinately: ae is to ai, as oi is to ou: wherefore if triangles be equiangled, they are proportional in shankes.

This converse is thus demonstrated. Let there be two triangles aei, and ouy, proportionall in shankes: And as ae is to ei; so let ou, be to uy: And as ai is to ie; so let oy bee to yu. Then at the points u and y, let angles be made by the [11 e iij]. equall to the angles at e and i, and let the triangle uys, be made: for the other angles at a and s, shall be equall by the [4 e]. And the triangle yus, shall be equiangled to the assigned aei. And by the antecedent, it shall be proportionall to it in shankes. Thus are two triangles ouy, by the grant; and uys, by the construction, proportionall in shanks to the same triangle aei: And as ae, is to ei, so is ou, to uy; so is su, to uy. Therefore seeing ou and su, are proportionall to the same yu, they are equall; Item, as ai is to ie: so is oy unto yu: so also is sy unto yu. Therefore oy and sy, seeing they are proportionall to the same yu, are equall. (yu is the common side.) The triangle therefore ouy, is equilater unto the triangle syu. And by the [1 e], it is to it equiangle: And therefore it is equiangled to the triangle aei, which was to be prooved. This was generally before taught at the [20 e iiij], of homologall sides subtending equall angles.

Therefore,

13. If a right line in a triangle be parallell to the base, it doth cut off from it a triangle equiangle to the whole, but lesse in base.

As in the triangle aei, the right line ou, doth cut off the triangle aou, equiangle, by the [21 e v], to the whole aei; But the base ou, is lesse than the base ei, as appeareth by the [21 e], and by the alternation of the sides.

14. If two trangles be proportionall in the shankes of the equall angle, they are equiangles: 6 p vj.

Let therefore the triangles given be aei, and ouy, equall in their angles a and o: And in their shankes let ea, be unto ai, as ou is to oy: And by the [11 e iij], let the angles soy, and oys, be equall to the angles eai, and eia: The other at s and e, shall be equall, by the [4 e]. Here thou seest that the triangle aei, is equiangle unto oys. Now, by the [12 e.] as ea is to ai: so is so to oy: and therefore, by the grant, so is uo to oy. Therefore seeing that uo, and os, are proportionall to oy, they are both equall. Lastly, if the common shanke oy bee added to both the shankes ou, and oy, are equall to the shankes so and oy. [But by the construction the angles oys and aie are equall. And, by the [4 e], the other at s and e are equall. Therefore the first triangle aei, is made equiangled to the third. Now seeing the second triangle uoy is to the third soy, equall in the shanks of the equall angle, it is to the same equilater, and by the [1 e], equiangled: Shon.] Wherefore the second triangle ouy shall likewise be equiangled to osy, the third: And therefore if

two triangles proportionall in shankes be equall in the angle of their shankes, they are equiangles.

15. If triangles proportionall in shankes, and alternly parallell, doe make an angle betweene them, their bases are but one right line continued. 32 p. vj.

Or thus: If being proportionall in their feet, and alternately parallels, they make an angle in the midst betweene them, they have their bases continued in a right line: H.

The cause is out of the [14 e v]. For they shall make on each side, with the falling line ai, two angles equall to two right angles.

Let the triangles aei and oiu, be proportionall in shanks: As ae is to ai, so let io be to ou: And let ea bee parallel to io: And ai to ou: Item, let them make the angle aio, betweene them, to wit, betweene their middle shankes ai, and oi, I say their bases ei, and iu, are but one right line continued. For seeing that by the grant ae, and oi, are parallels: Item ai and uo, the right line ai and oi, shall make, by the [21 e v], the angles at a, and o, equall to the alterne angle aio: And therefore they are equall betweene themselves: And then, by the [14 e], the triangles given are equiangles: Therefore the angle oui, is equall to the angle aie: Wherfore the three angles oiu, oia, and aie, by the [3 e], are equall to the three angles of the triangle eai, which are equall by the [13 e vj]. Unto two right angles: And therefore they themselves also are equall to two right angles. Wherefore, by the [14 e v], ei, and iu, are one right line continued.

16. If two triangles have one angle equall, another proportionall in shankes, the third homogeneall, they are equiangles. 7. p. vj.

Let aei, and ouy, the triangles given be equall in their angles a, and o: and proportionall in the shankes of the angles e, and u: and their other angles, at i, and y, homogeneall, that is, let them be both, either acute, or obtuse, or right angles. But first let them be acute, I say, the other at e, & u, are equall. Otherwise let aes, by the [11 e iij.] be made equall to the same ouy; Then have you them by the [4 e], equiangles; and the angles ase, shall be equall to the angle oyu; and both are acute angles: and by the [12. e], aes, and ouy, are proportionall in sides: and as ae, is to es; so shall ou, be to uy, that is, by the grant, so shall ae, be to ei. Therefore because the same ea, hath unto two, to wit, es, and ei, the same reason, the said es, and ei, are equall one to another: And therefore, by the [17. e. vj.] the angles at the base in s and i, are equall. Therefore both of them are acute angles: And in like manner ase, is an acute angle, contrary to the [14. e v]. The same will fall out altogether like to both the other, being either obtuse or right angles. The last part of a right angle is manifest by the [4 e] of this Booke.

Of Geometry the eight Booke, of the diverse kindes of Triangles.

1. A triangle is either right angled, or obliquangled.

The division of a triangle, taken from the angles, out of their common differences, I meane, doth now follow. But here first a speciall division, and that of great moment, as hereafter shall be in quadrangles and prismes.

2. A right angled triangle is that which hath one right angle: An obliquangled is that which hath none. 27. d j.

A right angled triangle in Geometry is of speciall use and force; and of the best Mathematicians it is called Magister matheseos, the master of the Mathematickes.

Therefore

3. If two perpendicular lines be knit together, they shall make a right angled triangle.

As here in aei. This construction and manner of making of a right angled triangle, is drawne out of the definition of a right angle. For right lines perpendicular are the makers of a right angle, as is manifest by the [13. e iij].

4. If the angle of a triangle at the base, be a right

angle, a perpendicular from the toppe shall be the other shanke: [and contrariwise Schon.]

As is manifest in the same example.

5. If a right angled triangle be equicrurall, each of the angles at the base is the halfe of a right angle: And contrariwise.

As in the triangle aei: For they are both equall to one right angle, by the [13. e. vj]. And betweene themselves, by the [17. e. vj].

Therefore

6. If one angle of a triangle be equall to the other two, it is a right angle [And contrariwise Schon.]

Because it is equall to the halfe of two right angles, by the [13. e vj].

And

7. If a right line from the toppe of a triangle cutting the base into two equall parts be equall to the bisegment, or halfe of the base, the angle at the toppe is a right angle: [And contrariwise Schon.]

As in the triangle aei, the right line ao, cutting the base ei, in o, into two equall parts, is equall to eo, or oi, the halfe of the base maketh two equicrural triangles; and the severall angles at the top equall to the angles at the ends, viz. e, and i, by the [17. e. vj]. Therefore the angle at the toppe

is equall to the other two: wherefore by the [6 e], it is a right angle.

8. A perpendicular in a triangle from the right angle to the base, doth cut it into two triangles, like unto the whole and betweene themselves, 8. p vj. [And contrariwise Schon.]

As in the triangle aei, the perpendicular ao, doth cut the triangles aoe, and aoi, like unto the whole aei, because they are equiangles to it; seeing that the right angle on each side is one, and another common in i, and e: Therefore the other is equall to the remainder, by [4. e vij]. Wherefore the particular triangles are equiangles to the whole: As proportionall in the shankes of the equall angles, by the [12. e vij]. But that they are like betweene themselves it is manifest by the [22. e iiij].

Therefore

9. The perpendicular is the meane proportionall betweene the segments or portions of the base.

As in the said example, as io, is to oa: so is oa, to oe, because the shankes of equall angles are proportionall, by the [8. e]. From hence was Platoes Mesographus invented.

And

10. Either of the shankes is proportionall betweene the base, and the segment of the base next adjoyning.

For as ei, is unto ia, in the whole triangle, so is ai, to io, in the greater. For so they are homologall sides, which

doe subtend equall angles, by the [23. e. iiij]. Item, as ie, is to ea; in the whole triangle, so is ae, to eo, in the lesser triangle.

Either of the shankes is proportionall betweene the summe, and the difference of the base and the other shanke. And contrariwise. If one side be proportionall betweene the summe and the difference of the others, the triangle given is a rectangle. M. H. Brigges.

This is a consectary arising likewise out of the [4 e.] of very great use.

In the triangle ead, the shanke ad, 12. is the meane proportionall betweene bd, 18. (the summe of the base ae, 13. and the shanke ed, 5.) and 8. the difference of the said base and shanke: For if thou shalt draw the right lines ba, and ac, the angle bac, shall be by the [6. e], a rectangle; (because it is equall to the angles at b, and c, seeing that the triangles bea, and eac, are equicrurall.) And by the [9 e], bd, da, and dc, are continually proportionall.

If a quadrate of a number, given for the first shanke, be divided of another, the halfe of the difference of the divisour, and quotient shall be the other shanke, and the halfe of the summe shall be the base. Or thus, The side of divided number doubled, and the difference of the divisour and quotient, shall be the two shankes, and the summe of them shall be the base.

Let the number given for the first shanke be 4. And let 8. divide 16. the quadrate of 4. by 2. The halfe of 8 - 2, that is 3. shall be the other shanke: And the halfe of 8 + 2, that is 5. shall be the base.

Therefore

If any one number shall divide the quadrate of another, the side of the divided, and the halfe of the difference of the divisour and the quotient, shall be the two shankes of a rectangled triangle, and the halfe of the summe of them shall be the base thereof.

Let the two numbers given be 4. and 6. The square of

6. let it be 36. and the quotient of 36. by 4. be 9: And the side is 6. for the one shanke. Now 9 - 4. that is, 5. is the difference of the divisour and quotient, whose halfe 2.½, is the other shanke. And 9 + 4. that is 13. is the summe the said devisour and quotient, whose halfe 6.½, is the base.

Againe let 4. and 8. be given. The quadrate of 8. is 64. And the quotient of 64 is 16. and the side of 64. is 8. for the one shanke. The halfe 16 - 4. that is 6. is the other shanke. And the halfe of 16 + 4. that is 10, is the base.

11. If the base of a triangle doe subtend a right-angle, the rectilineall fitted to it, shall be equall to the like rectilinealls in like manner fitted to the shankes thereof: And contrariwise, out of the 31. p. vj.

Or thus: If the base of a triangle doe subtend a right angle, the right lined figure made upon the base, is equall to the right lined figures like, and in like manner situate upon the feete: H.

Let the right angled triangle be aei: and let there be also the triangles eau, and aiy, and to them upon the base of the said right angle, by the [23 e iiij]. let the triangle ies, be made like, and in like manner situate. I say, that eis, is equall joyntly to eau, and aiy. Let ao, a perpendicular fall from the right angle a, to the base ei: This by the ioe, doth yeeld us twise three proportionals, to wit, ie, ea, eo: Item, ei, ia, io: Therefore, by the [25. e. iiij], as ie, is to eo: so is the triangle ies, to the triangle eau; And as ei, is to oi, so is the triangle eis, to the triangle aiy: But ei, is equall to eo, and oi, the whole, to wit, to his parts. Wherefore by the second composition in

Arithmeticke (9. c. ij.) the triangle eis, is equall to the triangles eau, and iay.

The Converse is thus proved: Let the triangle be aei: And let the perpendicular eo, be erected upon ae, equall to ei: And draw a right line from o to a: Here by the former, the rectilinealls situate at oe, and ea, that is by the construction, at ae, and ie, are equall to the rightilineall at ao, made alike and situate alike: And by the graunt they are equall, to the rectilineall at ai, made alike and situated alike. Therefore seeing the like rectilineals at ao, and ai, are equall; they have by the [20 e iiij], their homologall sides equall: And the two triangles are equiliters: And by the [1 e vij], equiangles. But aeo, is a right angle, by the construction: And aei, is proved to be equall to the same aeo: Therefore, by the [13 e v]. aei, also is a right angle.

12. An obliquangled triangle is either Obtusangled or Acutangled.

The division of an obliquangled triangle is taken from the speciall differences of an oblique angle. For at the 15 e iij, we were taught that an oblique angle was either obtuse or acute: Therefore an obliquangled triangle is an obtuseangle, and an Acutangle.

13. An obtusangle is that triangle which hath one blunt corner. 28.d i.

There can be but one right angle in a triangle, by the [2 e]. Therefore also in it there can be but one blunt angle.

Therefore

14. If the obtuse or blunt angle be at the base of the triangle given, a perpendicular drawne from the toppe

of the triangle, shall fall without the figure: And contrarywise.

As here in aei, the perpendicular io, falleth without: This is manifest by the [4 e].

And

15. If one angle of a triangle be greater than both the other two, it is an obtuse angle: And contrariwise.

This is plaine by the [6 e].

And

16. If a right line drawne from the toppe of the triangle cutting the base into two equall parts, be lesse than one of those halfes, the angle at the toppe is a blunt-angle. And contrariwise.

As in aei, the perpendicular eo, cutting the base ai into two equall parts ao, and oi: And the said eo is lesse than either ao, or oi: Therefore the angle aei, is a blunt angle by the [7 e].

17. An acutangled triangle is that which hath all the angles acute. 29 d j.

Therefore

18. A perpendicular drawne from the top falleth within the figure: And contrariwise.

As in aei, the perpendicular ao falleth within as is plaine by the [4 e].

And

19. If any one angle of triangle be lesse then the other two, it is acute: And contrariwise.

As is manifest by the [6 e].

And

20. If a right line drawne from the toppe of the triangle; cutting the base into two equall parts, be greater than either of those portions, the angle at the toppe is an acute angle: And contrariwise.

As in aei, let ao cutting the base ei into two equall parts, be greater than any one of those parts, the angle at the toppe is an acuteangle, as appeareth by the [7 e].

The ninth Booke, of P. Ramus Geometry, which intreateth of the measuring of right lines by like right-angled triangles.

The Geometry of like right-angled triangles, amongst many other uses that it hath, it doth especially afford us the geodæsy or measuring of right lines: And that mastery, which before (at the [2 e viij]) attributed the right angled triangles, shall here be found to be a true mastery indeed.

For it shall containe the geodesy of right lines; and afterward the geodesy of plaines and solides, by the measuring of their sides, which are right lines.

1. For the measuring of right lines; we will use the Iacobs staffe, which is a squire of unequall shankes.

Radius, commonly called Baculus Iacob, Iacobs staffe, as if it had been long since invented and practised by that holy Patriarke, is a very auncient instrument, and of all other Geometricall instruments, commonly used, the best and fittest for this use. Archimedes in his book of the Number of the sand, seemeth to mention some such thing: And Hipparchus, with an instrument not much unlike this, boldly attempted an haynous matter in the sight of God, as Pliny thinketh, namely to deliver unto posterity the number of the starres, and to assigne or fixe them in their true places by the Norma, the squire or Iacobs staffe. And indeed true it is that the Radius is not onely used for the measuring of the earth and land: But especially for the defining or limiting of the starres in their places and order: And for the describing and setting out of all the regions and waies of the heavenly city. Yea and Virgill the famous Poet, in his 3 Ecloge, Ecquis fuit alter, Descripsit radio totum, qui gentibus orbem? and againe afterward in the 6 of his Eneiades, hath noted both these uses. Cœliquè meatus. Describent radio & surgentia sidera dicent. Long after this the Iewes and Arabians, as Rabbi Levi; But in these latter daies, the Germaines especially, as Regiomontanus; Werner, Schoner, and Appian have grac'd it: But above all other the learned Gemma Phrisius in a severall worke of that argument onely, hath illustrated and taught the use of it plainely and fully.

The Iacobs staffe therefore according to his owne, and those Geometricall parts, shall here be described (The

astronomicall distribution wee reserve to his time and place.) And that done, the use of it shall be shewed in the measuring of lines.

This instrument, at the discretion of the measurer may be greater or lesser. For the quantity of the same can no otherwayes be determined.

2. The shankes of the staffe are the Index and the Transome.

The principall parts of this instrument are two, the Index, or Staffe, which is the greater or longer part: and the Transversarium, or Transome, and is the lesser and shorter.

3. The Index is the double and one tenth part of the transome.

Or thus: The Index is to the transversary double and 1/10 part thereof. H. As here thou seest.

4. The Transome is that which rideth upon the Index, and is to be slid higher or lower at pleasure.

Or, The transversary is to be moved upon the Index, sometimes higher, sometimes lower: H. This proportion in defining and making of the shankes of the instrument is perpetually to be observed: as if the transome be 10. parts, the

Index must be 21. If that be 189. this shall be 90. or if it be 2000. this shall be 4200. Neither doth it skill what the numbers be, so this be their proportion. More than this, That the greater the numbers be, that is the lesser that the divisions be, the better will it be in the use. And because the Index must beare, and the transome is to be borne; let the index be thicker, and the transome the thinner.

But of what matter each part of the staffe be made, whether of brasse or wood it skilleth not, so it be firme, and will not cast or warpe. Notwithstanding, the transome will more conveniently be moved up and downe by brasen pipes, both by it selfe, and upon the Index higher or lower right angle wise, so touching one another, that the alterne mouth of the one may touch the side of the other. The thrid pipe is to be moved or slid up and downe, from one end of the transome to the other; and therefore it may be called the Cursor. The fourth and fifth pipes, fixed and immoveable, are set upon the ends of the transome, are

unto the third and second of equall height with finnes, to restraine when neede is, the opticke line, and as it were, with certaine points to define it in the transome.

The three first pipes may, as occasion shall require, be fastened or staied with brasen scrues. With these pipes therefore the transome may be made as great, as need shall require, as here thou seest.

The fabricke or manner of making the instrument hath hitherto beene taught, the use thereof followeth: unto which in generall is required: First, a just distance. For the sight is not infinite. Secondly, that one eye be closed: For the optick faculty conveighed from both the eyes into one, doth aime more certainely; and the instrument is more fitly applied and set to the cheeke bone, then to any other place. For here the eye is as it were the center of the circle, into which the transome is inscribed. Thirdly, the hands must be steady; for if they shake, the proportion of the Geodesy must needes be troubled and uncertaine. Lastly, the place of the station is from the midst of the foote.

5. If the sight doe passe from the beginning of one shanke, it passeth by the end of the other: And the one shanke is perpendicular unto the magnitude to be measured, the other parallell.

These common and generall things are premised. That the sight is from the beginning of the Index by the end of the transome; Or contrariwise, From the beginning of the transome, unto the end of the Index. And that the Index is right, that is, perpendicular to the line to be measured, the transome parallell. Or contrariwise. Now the perpendicularity of the Index, in measurings of lengthts, may be tried by a plummet of lead appendent; But in heights and breadths, the eye must be trusted; although a little varying of the plummet can make no sensible errour.

By the end of the transome, understand that which is made by the line visuall, whether it be the outmost finne, or the Cursour in any other place whatsoever.

6. Length and Altitude have a threefold measure; The first and second kinde of measure require but one distance, and that by granting a dimension of one of them, for the third proportionall: The third two distances, and such onely is the dimension of Latitude.

Geodesy of right lines is two fold; of one distance, or of two. Geodesy of one distance is when the measurer for the finding of the desired dimension doth not change his place of standing. Geodesy of two distances is when the measurer by reason of some impediment lying in the way betweene him and the magnitude to be measured, is constrained to change his place, and make a double standing.

Here observe, That length and heighth, may be joyntly measured both with one, and with a double station: But breadth may not be measured otherwise than with two.

7. If the sight be from the beginning of the Index right or plumbe unto the length, and unto the farther end of the same, as the segment of the Index is, unto the segment of the transome, so is the heighth of the measurer unto the length.

Let therefore the segment of the Index, from the toppe, I meane, unto the transome be 6. parts. The segment of the transome, to wit, from the Index unto the opticke line be 18. The Index, which here is the heighth of the measurer, 4. foote: The length, by the rule of three, shall be 12. foote. The figure is thus, for as ae, is to ei, so is ao,

unto ou, by the [12. e vij]. For they are like triangles. For aei, and aou, are right angles: And that which is at a is common to them both: Wherefore the remainder is equall to the remainder, by the [4. e vij].

The same manner of measuring shall be used from an higher place; as out of y, the segment of the Index is 5. parts; the segment of the transome 6: and then the height be 10 foote: the same Length shall be found to bee 12 foote.

Neither is it any matter at all, whether the length in a plaine or levell underneath: Or in an ascent or descent of a mountaine, as in the figure under written.

Thus mayest thou measure the breadths of Rivers, Valleys, and Ditches. For the Length is alwayes after this manner, so that one may measure the distance of shippes on the Sea, as also Thales Milesius, in Proclus at the 26 p j, did measure them. An example thou hast here.

Hereafter in the measuring of Longitude and Altitude, sight is unto the toppe of the heighth. Which here I doe now forewarne thee of, least afterward it should in vaine be reitered often.

The second manner of measuring a Length is thus:

8. If the sight be from the beginning of the index parallell to the length to be measured, as the segment of the transome is, unto the segment of the index, so shall the heighth given be to the length.

As if the segment of the Transome be 120 parts: the height given 400 foote: The segment of the Index 210 parts: The length, by the golden rule shall be 700 foote. The figure is thus. And the demonstration is like unto the former; or indeed more easier. For the triangles are equiangles, as afore. Therefore as ou is to ua: so is ei to ia.

This is the first and second kinde of measuring of a Longitude, by one single distance or station: The third which is by a double distance doth now follow. Here the transome, if there be roome enough for the measurer to goe farre enough backe, must be put lower, in the second distance.

9. If the sight be from the beginning of the

transverie parallell to the length to be measured, as in the index the difference of the greater segment is unto the lesser; so is the difference of the second station unto the length.

This kinde of Geodæsy is somewhat more subtile than the former were. The figure is thus; in which let the first ayming be from a, the beginning of the transome, and out of ai the length sought by o, the end of the Index, unto e, the toppe of the heighth: And let the segment of the Index be ou: The second ayming let it be from y, the beginning of the transome, out of a greater distance by s, the end of the Index, unto e, the same note of the heighth: And let the segment of the Index be sr.

Here the measuring performed, is the taking of the difference betweene ou and sr. The rest are faigned onely for demonstrations sake. Therefore in the first station let aml, be from the beginning of the transome, be parallell to ye. Here first mu, is equall to sr. For the triangles

mua, and sry, are equall in their shankes ua, and ry, by the grant (Because the transome standeth still in his owne place:) And the angles at mua, uam, are equall to the angles: And all right angles are equall, by the [14 e iij]. These are the outter and inner opposite one to another: And such are equall by the 1 e v. Therefore they are equilaters, by the [2 e vij]; And om, is the difference of the segments of the Index. Then as om is to mu, so is el, to li; as the equation of three degrees doth shew. For, by the [12 e vij], as om is to ma: so is el to la: And as ma is to mu; so is la, to li. Therefore by right, as om, is to mu: so is el, to li: And by the [12 e vj], so is ya, to ai: As if the difference of the first segment be 36 parts: The second segment be 72 parts: The difference of the second station 40 foote. The length sought shall be 80 foote. And here indeed is no heighth definitely given, that may make any bound of the principall proportion. Notwithstanding the Heighth, although it be of an unknowne measure, is the bound of the length sought: And therefore it is an helpe and meanes to argue the question. Because it is conceived to stand plumbe upon the outmost end of the length.

Therefore that third kinde of measuring of length is oftentimes necessary, when by neither of the former wayes the length may possibly be taken, by reason of some impediment in the way, to wit of a wall, or tree, or house, or mountaine, whereby the end of the length may not be seene, which was the first way: Nor an height next adjoyning to the end of the length is given, which is the second way.

Hitherto we have spoken of the threefold measure of longitude, the first and second out of an heighth given the third cut of a double distance: The measuring of heighth followeth next, and that is also threefold. Now heighth is a perpendicular line falling from the toppe of the magnitude, unto the ground or plaine whereon the measurer doth stand, after which manner Altitude or

heighth was defined at the [9 e iiij]. The first geodesy or manner of measuring of heighths is thus.

10. If the sight be from the beginning of the transome perpendicular unto the height to be measured, as the segment of the transome, is unto the segment of the Index, so shall the length given be to the height.

Let the segment of the transome be 60 parts: the segment of the Index 36: the Length given 120 foote: the height sought shall be, by the golden rule, 72 foote.

The Figure is thus: And the demonstration is by the [12 e vij], as afore: but here is to be added the height of the measurer; which if it be 4 foot, the whole height shall be 76 foote.

Therefore in an eversed altitude

11. If the sight be from the beginning of the Index parallell to the height, as the segment of the transome

is, unto the segment of the index, so shall the length given be, unto the height sought.

Eversa altitudo, An eversed altitude (Reversed, H:) is that which we call depth, which indeed is nothing else, in the Geometers sense, but heighth turned topsie turvie, as we say, or with the heeles upward. For out of the heighth concluded by subducting that which is above ground, the heighth or depth of a Well shall remaine.

Let the segment of the transome ae, be 5 parts: the segment of the Index ei, be 13: the diameter of the Well (which now standeth for the length:) be 10 foote, which at toppe is supposed to be equall to that at bottome: the opposite height, by the [12 e vij], and the golden rule shall

be 26 foote: From whence you must take the segment of the Index reaching over the mouth of the Well: And the true height (or depth) shall remaine; as if that segment of 13 parts be as much as 2 foote, the height sought shall be 24 foote. The second manner of measuring of heights followeth.

12. If the sight be from the beginning of the Index perpendicular to the heighth to be measured, as the segment of the Index is unto the segment of the Transome, so shall the length given be to the heighth.