Chapter XIII: Of Analogism, or the Same Proportion (3)

18. A solid angle may be conceived two ways. First, for the aggregate of all the angles, which are made by the motion of a strait line, while one extreme point thereof remaining fixed, it is carried about any plain figure, in which the fixed point of the strait line is not contained. And in this sense, it seems to be understood by Euclid. Now it is manifest, that the quantity of a solid angle so conceived is no other, than the aggregate of all the angles in a superficies so described, that is, in the superficies of a pyramidal solid. Secondly, when a pyramis or cone has its vertex in the centre of a sphere, a solid angle may be understood to be the proportion of a spherical superficies subtending that vertex to the whole superficies of the sphere. In which sense, solid angles are to one another as the spherical bases of solids, which have their vertex in the centre of the same sphere.

[Sidenote: What is the nature of asymptotes.]

19. All the ways, by which two lines respect one another, or all the variety of their position, may be comprehended under four kinds; for any two lines whatsoever are either parallels, or being produced, if need be, or moved one of them to the other parallelly to itself, they make an angle; or else, by the like production and motion, they touch one another; or lastly, they are _asymptotes_. The nature of parallels, angles, and tangents, has been already declared. It remains that I speak briefly of the nature of asymptotes.

Asymptosy depends upon this, that quantity is infinitely divisible. And from hence it follows, that any line being given, and a body supposed to be moved from one extreme thereof towards the other, it is possible, by taking degrees of velocity always less and less, in such proportion as the parts of the line are made less by continual division, that the same body may be always moved forwards in that line, and yet never reach the end of it. For it is manifest, that if any strait line, as A F, (in the 8th figure) be cut anywhere in B, and again B F be cut in C, and C F in D, and D F in E, and so eternally, and there be drawn from the point F, the strait line F F at any angle A F F; and lastly, if the strait lines A F, B F, C F, D F, E F, &c., having the same proportion to one another with the segments of the line A F, be set in order and parallel to the same A F, the crooked line A B C D E, and the strait line F F, will be _asymptotes_, that is, they will always come nearer and nearer together, but never touch one another. Now, because any line may be cut eternally according to the proportions which the segments have to one another, therefore the divers kinds of asymptotes are infinite in number, and not necessary to be further spoken of in this place. In the nature of asymptotes in general there is no more, than that they come still nearer and nearer, but never touch. But in special in the asymptosy of hyperbolic lines, it is understood they should approach to a distance less than any given quantity.

[Sidenote: Situation, by what it is determined.]

20. SITUATION _is the relation of one place to another_; and where there are many places, their situation is determined by four things; by _their distances from one another_; by _several distances from a place assigned_; by _the order of strait lines drawn from a place assigned to the places of them all_; and by _the angles which are made by the lines so drawn_. For if their distances, order, and angles, be given, that is, be certainly known, their several places will also be so certainly known, as that they can be no other.

[Sidenote: What is like situation; what is figure; and what are like
figures.]

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.

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_Vol. 1. Lat. & Eng._
C. XIV.
_Fig. 1-10_
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