21. Points, how many soever they be, have like situation with an equal number of other points, when all the strait lines, that are drawn from some one point to all these, have severally the same proportion to those, that are drawn in the same order and at equal angles from some one point to all those. For let there be any number of points as A, B, and C, (in the [9th figure]) to which from some one point D let the strait lines D A, D B, and D C be drawn; and let there be an equal number of other points, as E, F, and G, and from some point H let the strait lines H E, H F, and H G be drawn, so that the angles A D B and B D C be severally and in the same order equal to the angles E H F and F H G, and the strait lines D A, D B, and D C proportional to the strait lines H E, H F, and H G; I say, the three points A, B, and C, have like situation with the three points E, F, and G, or are placed alike. For if H E be understood to be laid upon D A, so that the point H be in D, the point F will be in the strait line D B, by reason of the equality of the angles A D B and E H F; and the point G will be in the strait line D C, by reason of the equality of the angles B D C and F H G; and the strait lines A B and E F, as also B C and F G, will be parallel, because A D. E H :: B D. F H :: C D. G H are proportionals by construction; and therefore the distances between the points A and B, and the points B and C, will be proportional to the distances between the points E and F, and the points F and G. Wherefore, in the situation of the points A, B, and C, and the situation of the points E, F and G, the angles in the same order are equal; so that their situations differ in nothing but the inequality of their distances from one another, and of their distances from the points D and H. Now, in both the orders of points, those inequalities are equal; for A B. B C :: E F. F G, which are their distances from one another, as also D A. D B. D C :: H E. H F. H G, which are their distances from the assumed points D and H, are proportionals. Their difference, therefore, consists solely in the magnitude of their distances. But, by the definition of like, (chapter I. [article 2]) those things, which differ only in magnitude, are like. Wherefore the points A, B, and C, have to one another like situation with the points E, F, and G, or are placed alike; which was to be proved.

Figure is quantity, determined by the situation or placing of all its extreme points. Now I call those points extreme, which are contiguous to the place which is without the figure. In lines therefore and superficies, all points may be called extreme; but in solids only those which are in the superficies that includes them.

Like figures are those, whose extreme points in one of them are all placed like all the extreme points in the other; for such figures differ in nothing but magnitude.

And like figures are alike placed, when in both of them the homologal strait lines, that is, the strait lines which connect the points which answer one another, are parallel, and have their proportional sides inclined the same way.

And seeing every strait line is like every other strait line, and every plane like every other plane, when nothing but planeness is considered; if the lines, which include planes, or the superficies, which include solids, have their proportions known, it will not be hard to know whether any figure be like or unlike to another propounded figure.

And thus much concerning the first grounds of philosophy. The next place belongs to geometry; in which the quantities of figures are sought out from the proportions of lines and angles. Wherefore it is necessary for him, that would study geometry, to know first what is the nature of quantity, proportion, angle and figure. Having therefore explained these in the three last chapters, I thought fit to add them to this part; and so pass to the next.

Vol. 1. Lat. & Eng.
C. XIV.
Fig. 1-10

[Fig 1.][Fig 2.][Fig 3.][Fig 4.][Fig 5.][Fig 6.][Fig 7.][Fig 8.][Fig 9.][Fig 10.]