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.
As if the segment of the Index be 60 parts: and the segment also of the transome be 60: And the Length given be 250 foote: By the Rule of three, the height also shall be 250 foote: as thou seest in the example underneath: For as ae is to ei; so is aeo to ou, by the [12 e vij]. But here unto the height found, you must adde the height of the measurer: Which if it be 4 foot, the whole height shall be 254 foote.
Therefore
13. If the sight be from the beginning of the Index (perpendicular to the magnitude to be measured) by the names of the transome, unto the ends of some known part of the height, as the distance of the Names is, unto the rest of the transome above them, so shall the known part be unto the part sought.
Or thus: If the sight passe from the beginning of the Index being right, by the vanes of the transversary, to the tearmes of some parts; as the distance of the vanes is unto the rest of the transversary above the index, so is the part knowne unto the remainder: H.
This is a consectary of a knowne part of an height, from whence the rest may be knowne, as in the figure.
As ou is unto uy, so is ei to is. For as ou, is unto ua: so is ei unto ia, by the [12 e vij]. And as ua, is to uy, so is ia unto is; and by right, as ou, is to uy, so is ei, to is. Here thou hast three bounds of the proportion. Let therefore ou, be 20 parts: uy 30: And ei, the knowne part, let it be
15 foote: Therefore thou shalt conclude is, the rest to be 22½.
The first and second kinde of measuring of heights is thus: The third followeth.
14 If the sight be from the beginning of the Index perpendicular to the heighth, as in the Index the difference of the segment, is unto the difference of the distance or station; so is the segment of the transome unto the heighth.
Hitherto you must recall that subtilty, which was used in the third manner of measuring of lengths.
Let the first aime be taken from a, the beginning of the Index perpendicular unto the height to be measured: And from an unknowne length ai, by o, the end of the transome, unto e, the toppe of the height ei: And let the segment of the Index be ua. The second ayme, let it be taken from y, the beginning of the same Index; and out of a
greater distance, by s, the end of the transome, unto the same toppe e. And the segment of the Index let it be ry.
Here, as afore, the measuring is performed and done, by the taking of the difference of the said yr, above au: Now the demonstration is concluded, as in the former was taught. Let the parallell lsm, be erected against aoe.
Here first the triangles oua, & srl, are equilaters, by the [2 e vij].; (seeing that the angles at a, and l, the externall and internall, are equall in bases ou, and sr, for the segment in each distance is the same still:) Therefore ua, is equall to rl. Now the rest is concluded by a sorites of foure degrees: As yr, is unto yi: so by the [12. e vij]. is sr, that is, ou, unto ei: And as ou, is unto ei, so is au, that is, lr, unto ai. Therefore the remainder yl, unto the remainder ya; shall be as yr, is unto the whole yi, and therefore from the first unto the last, as sr, is to ei.
Therefore let the difference of the Index be 23. parts: The difference of the distance 30. foote: The segment of the transome 44. parts: The height shall be 57.9/23. or foote.
Therefore
15 Out of the Geodesy of heights, the difference of two heights is manifest.
Or thus: By the measure of one altitude, we may know the difference of two altitudes: H.
For when thou hast taken or found both of them, by some one of the former wayes, take the lesser out of the greater; and the remaine shall be the heighth desired. From hence therefore by one of the towers of unequall heighth, you may measure the heighth of the other. First out of the lesser, let the length be taken by the first way: Because the height of the lesser, wherein thou art, is easie to be taken, either by a plumbe-line, let fall from the toppe to the bottom, or by some one of the former waies. Then measure
the heighth, which is above the lesser: And adde that to the lesser, and thou shalt have the whole heighth, by the first or second way. The figure is thus, and the demonstration is out of the [12. e vij]. For as ae, is to ei, so is ao, to ou. Contrariwise out of an higher Tower, one may measure a lesser.
16 If the sight be first from the toppe, then againe from the base or middle place of the greater, by the vanes of the transome unto the toppe of the lesser heighth; as the said parts of the yards are unto the part of the first yard; so the heighth betweene the stations shall be unto his excesse above the heighth desired.
Let the unequall heights be these, as, the lesser, and uy, the greater: And out of the assigned greater uy, let the lesser, as, be sought. And let the sight be first from u, the toppe of the greater, unto a, the toppe of the lesser,
making at the shankes of the staffe the triangle urm. Then againe let the same sight be from the base, or from the lower end of uy, the heighth given, unto a, the same toppe of the lesser, making by the shankes of the staffe the triangle yln, so that the segments of the yard be, the upper one, I meane, ur, the neather one ul: I say the whole of ur, and nl, is unto ur: so is the uy, greater heighth assigned, unto as, the lesser sought.
The Demonstration, by drawing of ao, a perpendicular unto uy, is a proportion out of two triangles of equall heighth. For the forth of the totall equally heighted triangles uao, and yas, although they be reciprocall in situation, they have their bases uo, and as, as if their were oy. Then they have the same with the whole triangles; as also the subducted triangles urm, and ynl, of equal heighth; to wit whose common heighth is the segment of the transome remained still in the same place, there rm, here yl. And therefore the bases of these, namely, the segments of the yards ur, and nl, have the same rate with uo, unto oy.
As therefore uo, is unto oy: so is, ur, unto nl. And backward, as nl, is to ur; so is, yo, unto ou, as here thou seest:
| nl,————ur: | yo,————ou. |
Therefore furthermore by composition of the Antecedent with the Consequent unto the Consequent, by the 5 c 9 ij. Arith. As nl, and ur, are unto ur: so are yo, and ou, unto ou, that is yu, unto ou, on this manner.
| nl,————ur, | yo,————ou, | |
| nr, | ou, | |
| ——————————————— | ||
| ... ur, | yu, ou, | |
there is given nl, and ur, for the first proportionall: ur, for the second: and yu, for the third: Therefore there is also given ou, for the fourth: Which ou, subducted out of uy, there remaineth oy, that is, as, the lesser altitude sought.
For let the parts of the yard be 12. and 6. and the summe of them 18. Now as 18. is unto 12. so is the whole altitude uy, 190. foote, unto the excesse 126⅔ foote. The remainder therefore 63⅓ foote, shall be as, the lesser heighth sought.
But thou maist more fitly dispose and order this proportion thus: As ur, is unto nl: so is uo unto oy. Therefore by Arithmeticall composition, as ur, and nl, are unto nl: so uo, and oy, that is, the whole uy, is unto oy, that is, unto as. For here a subduction of the proportion, after the composition is no way necessary, by the crosse rule of societia, thus:
The second station might have beene in o, the end of the perpendicular from a. But by taking the ayme out of the toppe of the lesser altitude, the demonstration shall be yet againe more easie and short, by the two triangles at the yard aei, and aef, resembling the two whole triangles aou, and aoy, in like situation, the parts of the
shanke cut, are on each side the segments of the transome.
One may againe also out of the toppe of a Turret measure the distance of two turrets one from another: For it is the first manner of measuring of longitudes, neither doth it here differ any whit from it, more than the yard is hang'd without the heighth given. The figure is thus: And the Demonstration is by the [12. e vij]. For as ae, the segment of the yard, is unto ei the segment of the transome: so is the assigned altitude ao, unto the length ou.
The geodesy or measuring of altitude is thus, where either the length, or some part of the length is given, as in the first and second way: Or where the distance is double, as in the third.
17 If the sight be from the beginning of the yard being right or perpendicular, by the vanes of the transome, unto the ends of the breadth; as in the yard the difference of the segment is unto the differēce of the distance, so is the distance of the vanes unto the breadth.
The measuring of breadth, that is of a thwart or crosse
line, remaineth. The Figure and Demonstration is thus: The first ayming, let it be aei, by o, and u, the vanes of the transome ou. The second, let it be yei, by s, and r, the vanes of the transome sr. Then by the point s, let the parallell lsm, be drawne against aoe. Here first, the triangles oua, and sil, are equilaters, by the [2 e vij]. Because the angles at n and j, are right angles: And uao, and jls, the outter and inner, are equall in their bases ou, and sj, by the grant: Because here the segment of the transome remaineth the same: Therefore ua, is equall to jl. These grounds thus laid, the demonstration of the third altitude here taken place. For as yl, is unto ya: so is sj, unto er: And, because parts are proportionall unto their multiplicants, so is sr, unto ei: for the rest doe agree.
The same shall be the geodesy or manner of measuring, if thou wouldest from some higher place, measure the breadth that is beneath thee, as in the last example. But from the distance of two places, that is, from latitude or breadth, as of Trees, Mountaines, Cities, Geographers and Chorographers do gaine great advantages and helpes.
Wherefore the geodesy or measuring of right lines is thus in length, heighth, and breadth, from whence the Painter, the Architect, and Cosmographer, may view and gather of many famous place the windowes, the statues or imagery, pyramides, signes, and lastly, the length and heighth, either by a single or double: the breadth by a double dimension onely, that is, they may thus behold and take of all places the nature and symmetry; as in the example next following thou mayst make triall when thou pleasest.
The tenth Booke of Geometry, of a Triangulate and Parallelogramme.
And thus much of the geodesy of right lines, by the meanes of rectangled triangles: It followeth now of the triangulate.
1. A triangulate is a rectilineall figure compounded of triangles.
As before (for the dichotomies sake) of a line was made a Lineate, to signifie the genus of surface and a Body: so now is for the same cause of a triangle made a Triangulate, to declare and expresse the genus of a Quadrilater and Multilater, and indeed more justly, then before in a Lineate. For triangles doe compound and make the triangulate, but lines doe not make the lineate.
Therefore
2. The sides of a triangulate are two more than are the triangles of which it is made.
As the sides of a Quadrangle are 4. Therefore the triangles which doe make the same foure-sided figure are but 2. The sides of a Quinquangle are 5, Therefore the triangles are 3, and so forth of the rest, as here thou seest. And
that indeed is the least: For even a triangle it selfe, may be cut into as many triangles as one please.
That both the inner and outter are equall to right angles, in every kinde of right line figure, it was manifest at the [4 e vj]. The inner in a Quadrangle, are equall to 4. In a Quinquangle, to 6: In an Hexangle, to 8; and so forth.
But the outter, in every right-lined figure, are equall to 4 right angles: as here may be demonstrated, by the [14 e v] and [13 e vj].
And
3. Homgeneall Triangulates are cut into an equall number of triangles, è 20 p vj.
For if they be Quadrangles, they be cut into two triangles: If Quinquangles, into 3. If Hexangles, into 4, and so forth.
4. Like triangulates are cut into triangles alike one to another and homologall to the whole è 20 p vj.
Or thus: Like Triangulates are divided into triangles like one unto another, and in porportion correspondent unto the whole: H.
As in these two quinquangles. First the particular triangles are like betweene themselves. For the shankes of aeu and ysm, equall angles are proportionall, by the grant. Therefore the triangles themselves are equiangles, by [14 e vij]. And therefore alike, by the [12 e vij], and so forth of the rest.
The middle triangles, the equall angles being substracted shall have their other angles equall: And therefore they also shall be equiangles and alike, by the same.
Secondarily, the triangles aeu. and ysm: eio and srl; eou, and slm, to wit, alike betweene themselves, are by the [1 e vj], in a double reason of their homologall sides eu, sm, eo, sl, which reason is the same, by meanes of the common sides. Therefore three triangles are in the same reason: And therefore they are proportionall: And, by the third composition, as one of the antecedents is, unto one of the consequents; so is the whole quinquangle to the whole.
5. A triangulate is a Quadrangle or a Multangle.
The parts of this partition are in Euclide, and yet without any shew of a division. And here also, as before, the species or severall kinds have their denomination their angles, although it had beene better and truer to have beene taken from their sides; as to have beene called a Quadrilater, or a Multilater. But in words use must bee followed as a master.
6. A Quadrangle is that which is comprehended of foure right lines. 22 d j.
As here thou seest. But a Quadrangle may also bee a sphearicall, and a conicall, and a cylindraceall, and that
those differences are common, we doe foretell at the [3 e v]. And a Quadrangle may be a plaine, which is not a quadrilater, as here.
7. A quadrangle is a Parallelogramme, or a Trapezium.
This division also in his parts is in the Elements of Euclide, but without any forme or shew of a division. But the difference of the parts shall more fitly be distinguished thus: Because in generall there are many common parallels.
8. A Parallelogramme is a quadrangle whose opposite sides are parallell.
As in the example, the side ae, is parallell to the side io: And the side ei, is parallell to opposite side ao.
Therefore
9. If right lines on one and the same side, doe joyntly bound equall and parallall lines, they shall make a parallelogramme.
The reason is, because they shall be equall and parallell betweene themselves, by the [26. e v].
And
10 A parallelogramme is equall both in his opposite sides, and angles, and segments cut by the diameter.
Or thus: The opposite, both sides, and angles, and segments cut by the diameter are equall. Three things are here concluded: The first is, that the opposite sides are equall: This manifest by the [26 e v]. Because two right lines doe jointly bound equall parallells.
The second, that the opposite angles are equall, the Diagonall ai, doth shew. For it maketh the triangles aei, and ioa, equilaters: And therefore also equiangles: And seeing that the particular angles at a, and i, are equall, the whole is equall to the whole. This part is the 34. p j;
The third: The segments cut by the diameter are alwayes equall, whether they be triangles, or any manner of quadrangles, as in the figures. For the Diameter doth cut into two equall parts, the parallelogramme by the Angles, or by the opposite sides, or by the alternall equall segments of the sides.
And
11. The Diameter of a parallelogramme is cut into two by equall raies.
As in the three figures aei, next before: This a parallelogramme hath common with a circle, as was manifest at the [28. e iiij].
And
12 A parallelogramme is the double of a triangle of a trinangle of equall base and heighth, 41. p j.
The comparison first in rate of inequality of a parallelogramme with a triangle, doth follow: As here thou seest in this diagramme. For a parallelogramme is cut into two equall triangles, by the antecedent. Therefore it is the double of the halfe.
And
13 A parallelogramme is equall to a triangle of equall heighth and double base unto it: è 42. p j.
As to aei, the triangle, the parallelogramme aoiu, is equall: because halfe of the parallelogramme is equall to the triangle: Therefore the halfes being equall, whole also shall be equall.
From whence one may
14 To a triangle given, in a rectilineall angle given, make an equall parallelogramme.
As here to the triangle, aei, given in s, the right lined angle given, you may equall the parallelogramme ouyi.
15 A parallelogramme doth consist both of two diagonals, and complements, and gnomons.
For these three parts of a parallelogramme are much used in Geometricall workes and businesses, and therefore they are to be defined.
16 The Diagonall is a particular parallelogramme having both an angle and diagonall diameter common with the whole parallelogramme.
First the Diagonall is defined: As in the whole parallelogramme aeio, the diagonals are auys, and ylir; Because they are parts of the whole, having both the same common angles at a, and i: and diagonall diameter ai, with the whole parallelogramme: Not that the whole diagonie is common to both: But because the particular diagonies are the parts of the whole diagony. Therefore the diagonalls are two.
17 The Diagonall is like, and alike situate to the whole parallelogramme: è 24. p vj.
There is not any, either rate or proportion of the diagonall propounded, onely similitude is attributed to it, as in the same figure, the Diagonall auys, is like unto the whole parallelogramme aeio. For first it is equianglar to it. For the angle at a, is common to them both: And that is equall to that which is at y, (by the [10. e x]:) And therefore also it is equall to that at i by the [10. e x]. Then the angles auy, and asy, are equall, by the [21. e v]. to the opposite inner angles at e, and o. Therefore it is equiangular unto it.
Againe, it is proportionall to it in the shankes of the
equall angles. For the triangles auy, and aei, are alike, by the [12 e vij], because uy is parallell to the base. Therefore as au is uy; so is ai to ei: Then as uy is to ya; so is ei to ia. Againe by the [21 e v], because sy is parallell to the base io, as ay is to ys: so is ai, to io: Therefore equiordinately, as uy is to ys: so is ei to io: Item as sy is to ya, so is io to ia: And as ya is to as: so is ia to ao. Therefore equiordinately, as ys is to sa: so is io to oa. Lastly as sa is unto ay; so is oa unto ai: And as ay is to au; so is ai unto ae. Therefore equiordinately, as sa is to au: so is ao, to ae. Wherefore the Diagonall su is proportionall in the shankes of equall angles to the parallelogramme oe.
The demonstration shall be the same of the Diagonall rl. The like situation is manifest, by the [21 e iiij]. And from hence also is manifest, That the diagonall of a Quadrate, is a Quadrate: Of an Oblong, an Oblong: Of a Rhombe, a Rhombe: Of a Rhomboides, a Rhomboides: because it is like unto the whole, and a like situate.
Now the Diagonalls seeing they are like unto the whole and a like situate, they shall also be like betweene themselves and alike situate one to another, by the [21] and [22 e iiij].
Therefore
18. If the particular parallelogramme have one and the same angle with the whole, be like and alike situate unto it, it is the Diagonall. 26 p vj.
This might have beene drawn, as a consectary, out of the former: But it may also as it is by Euclide be forced, by an argument ab impossibili. For otherwise the whole should be equall to the part, which is impossible.
As for example, Let the particular parallelogramme auys, be
coangular to the whole parallelogramme aeio; And let it have the same angle with it at a; like unto the whole and alike situate unto it; I say it is the Diagonall.
Otherwise, let the diverse Diagony be aro: And let lr be parallell against ae: Therefore alrs, shall bee the Diagonall, by the 6 e [[16].] Now therefore it shall be, by 8 e [[17 e,]] as ea is to ai: so is sa unto al: Againe,by the grant, as ea is unto ai: so is sa to au: Therefore the same sa is proportionall to al, and to au: And al is equall to au, the part to the whole, which is impossible.
19. The Complement is a particular parallelogramme, comprehended of the conterminall sides of the diagonals.
Or thus: It is a particular parallelogramme conteined under the next adjoyning sides of the diagonals.
As in this figure, are ur, and sy: For each of them is comprehended of the continued sides of the two diagonals. And therefore are they called Complements, because they doe with the Diagonals complere, that is, fill or make up the whole parallelogramme. Neither in deed may the two diagonals be described, but withall the complements must needes be described.
20. The complements are equall. 43 p j.
As in the same figure, are the sayd ur, and sr: For the triangles aei, and aoi, are equall, by the [12 e]. Item, so are asl, and aul: Item, so are lui, and lri. Therefore if you shall on each side take away equall triangles from those which are
equall, you shall leave the Complements equall betweene themselves.
Therefore
21. If one of the Complements be made equall to a triangle given, in a right-lined angle given, the other made upon a right line given shall be in like manner equall to the same triangle. 44 p j.
As if thou shouldest desire to have a parallelogramme upon a right line given, and in a right lined angle given, to be made equall to a triangle given, this proposition shall give satisfaction.
Let aei be the Triangle given: The Angle be o: And the right line given be iu: And the Parallelogramme ay is equall to aei, triangle given in the angle assigned, by the [13 e]. Then let the side ay, bee continued to r, equally to iu, the line given: And let ru be knit by a right line: And from r drawne out a diagony untill it doe meete with as, infinitely continued; which shall meete with it, by the [19 e v], in l. And the sides yi, and ru, let them be continued equally to sl. in m and n. And knit ln together with a right line. This complement mu, is equall to the complement ys, which is equall to the Triangle assigned, by the former, and that in a right lined angle given.
And
22 If parallelogrammes be continually made equall to all the triangles of an assigned triangulate, in a right lined angle given, the whole parallelogramme shall in like manner be equall to the whole triangulate. 45 p j.
This is a corollary of the former, of the Reason or rate of a Parallelogramme with a Triangulate; and it needeth no
farther demonstration; but a ready and steddy hand in describing and working of it.
Take therefore an infinite right line; upon the continue the particular parallelogrammes, As if the Triangulate aeiou, were given to be brought into a parallelogramme: Let it be resolved into three triangles, aei, aio, and aou: And let the Angle be y: First in the assigned Angle, upon the Infinite right line, make by the former the Parallelogramme ae, in the angle assigned, equall to aei, the first triangle. Then the second triangle, thou shalt so make upon the said Infinite line, that one of the shankes may fall upon the side of the equall complement; The other be cast on forward, and so forth in more, if neede be.
Here thou hast 3 complements continued, and continuing the Parallelogramme: But it is best in making and working of them, to put out the former, and one of the sides of the inferiour or latter Diagonall, least the confusion of lines doe hinder or trouble thee.
Therefore
23. A Parallelogramme is equall to his diagonals and complements.
For a Parallelogramme doth consist of two diagonals, and as many complements: Wherefore a Parallelogramme is equall to his parts: And againe the parts are equall to their whole.
24. The Gnomon is any one of the Diagonall with the two complements.
There is therefore in every Parallelogramme a double Gnomon; as in these two examples. Of all the space of a
parallelogramme about his diameter, any parallelogramme with the two complements, let it be called the Gnomon. Therefore the gnomon is compounded, or made of both the kindes of diagonall and complements.
In the Elements of Geometry there is no other use, as it seemeth of the gnomons than that in one word three parts of a parallelogramme might be signified and called by three letters aei. Otherwise gnomon is a perpendicular.
25. Parallelogrames of equall height are one to another as their bases are. 1 p vj.
As is apparent, by the [16 e iiij]. Because they be the double of Triangles, by the [12 e], of first figures: As ae, and ei.
Therefore
26 Parallelogrammes of equall height upon equall bases are equall. 35. 36 pj.
As is manifest in the same example.
27 If equiangle parallelogrammes be reciprocall in the shankes of the equall angle, they are equall: And contrariwise. 15 p vj.
It is a consectary drawne out of the [11 e vij]: As here thou seest: And yet indeed both that (as there was sayd) and this is rather a consectary of the [18 e iiij], which here also is more manifest.
Therefore
28 If foure right lines be proportionall, the parallelogramme made of the two middle ones, is equall to the equiangled parallelogramme made of the first and last: And contrariwise, e 16 p vj.
For they shall be equiangled parallelogrammes reciprocall in the shankes of the equall angle.
And
29 If three right lines be proportionall, the parallelogramme of the middle one is equall to the equiangled parallelogramme of the extremes: And contrariwise.
It is a consectary drawne out of the former.
Of Geometry, the eleventh Booke, of a Right angle.
1. A Parallelogramme is a Right angle or an Obliquangle.
Hitherto we have spoken of certaine common and generall matters belonging unto parallelogrammes: specials doe follow in Rectangles and Obliquangles, which difference, as is aforesaid, is common to triangles and triangulates. But at this time we finde no fitter words whereby to distinguish the generals.
2. A Right angle is a parallelogramme that hath all his angles right angles.
As in aeio. And here hence you must understand by one right angle that all are right angles. For the right angle at a, is equall to the opposite angle at i, by the [10 e x].
And therefore they are both right angles, by the [14 e iij]. The other angle at e, and o, by the [4 e vj], are equall to two right angles: And they are equall betweene themselves, by the [10 e x]. Therefore all of them are right angles. Neither
may it indeed possible be, that in a parallelogramme there should be one right angle, but by and by they must be all right angles.
Therefore
3 A rightangle is comprehended of two right lines comprehending the right angle 1. d ij.
Comprehension, in this place doth signifie a certaine kind of Geometricall multiplication. For as of two numbers multiplied betweene themselves there is made a number: so of two sides (ductis) driven together, a right angle is made: And yet every right angle is not rationall, as before was manifest, at the [12. e iiij], and shall after appeare at the [9 e].
And
4 Foure right angles doe fill a place.
Neither is it any matter at all whether the foure rectangles be equall, or unequall; equilaters, or unequilaters; homogeneals, or heterogenealls. For which way so ever they be turned, the angles shall be right angles: And therefore they shall fill a place.
5 If the diameter doe cut the side of a right angle into two aquall parts, it doth cut it perpendicularly: And contrariwise.
As here appeareth by the [1 e vij]. by drawing of the diagonies of the bisegments. The converse is manifest, by the [2 e vij]. and 17. e vij.
Therefore
6 If an inscribed right line doe perpendicularly cut the side of the right angle into two equall parts, it is the diameter.
The reason is, because it doth cut the parallelogramme into two equall portions.
7 A right angle is equall to the rightangles
made of one of his sides and the segments of the other.
As here the foure particular right angles are equall to the whole, which are made of ae, one of his sides, and of ei, io, ou, uy, the segments of the other.
The Demonstration of this is from the rule of congruency: Because the whole agreeth to all his parts. But the same reason in numbers is more apparent by an induction of the parts: as foure times eight are 32. I breake or divide 8. into 5. and 3. Now foure times 5. are 20. And foure times 3. are 12. And 20. and 12. are 32. And 32. and 32. are equall. Therefore 20. and 12. are also equall to 32.
Lastly, every arithmeticall multiplication of the whole numbers doth make the same product, that the multiplication of the one of the whole numbers given, by the parts of the other shall make: yea, that the multiplication of the parts by the parts shall make. This proportion is cited by Ptolomey in the 9. Chapter of the 1 booke of his Almagest.
8 If foure right lines be proportionall, the rectangle of the two middle ones, is equall to the rectangle of the two extremes. 16. p vj.
It is a speciall consectary out of the [28 e x]. As here are foure right lines proportionall betweene themselves: And the rectangle of the extremes, or first and last let it be ay: Of the middle ones, let it be se.
9 The figurate of a rationall rectangle is called a rectinall plaine. 16. d vij.
A rationall figure was defined at the [12. e iiij]. of which
sort amongst all the rectilineals hitherto spoken of, we have not had one: The first is a Right angled parallelogramme; And yet not every one indifferently: But that onely whose base is rationall to the highest: And that reason of the base and heighth is expressable by a number, where also the Figurate is defined. A rectangle or irrational sides, such as were mentioned at the [9 e j]. is irrationall. Therefore a rectangled rationall of rationall sides, is here understood: And the figurate thereof, is called, by the generall name, A Plaine: Because of all the kindes of Plaines, this kinde onely is rationall.
If therefore the Base of a Rectangle be 6. And the height 4. The plot or content shall be 24. And if it be certaine that the rectangles content be 24. And the base be 6. It shall also be certaine that the heighth is 4. The example is thus.
And this multiplication, as appeared at the [13. e iiij]. is geometricall: As if thou dost multiply 5. by 8. thou makest 40. for the Plaine: And the sides of this Plaine, are 5. and 8. it is all one as if thou hadst made a rectangled parallellogramme of 40. square foote content, whose base should be 5. foote, and the heigth 8. after this manner.
This manner of multiplication, say I, is Geometricall: Neither are there here, of lines made lines, as there of unities were made unities; but a magnitude one degree higher, to wit, a surface, is here made.
Here hence is the Geodesy or manner of measuring of a rectangled triangle made knowne unto us. For when thou shalt multiply the shankes of a right angle, the one by the other, thou dost make the whole rectangled parallelogramme, whose halfe is a triangle, by the [12. e x].