Chapter XIII: Of Analogism, or the Same Proportion (2)

23. If there be four proportionals, and a mean be interposed betwixt the first and second, and another betwixt the third and fourth, the first of these means will be to the second, as the first of the proportionals is to the third, or as the second of them is to the fourth. For let A. B :: C. D be proportionals, and let E be a mean betwixt A and B, and F a mean betwixt C and D; I say A. C :: E. F are proportionals. For the proportion of A to E is subduplicate of the proportion of A to B, or of C to D. Also, the proportion of C to F is subduplicate of that of C to D; and therefore A. E :: C. F are proportionals; and by permutation A. C :: E. F are also proportionals; which was to be proved.

24. Any thing is said to be divided into extreme and mean proportion, when the whole and the parts are in continual proportion. As for example, when A + B. A. B are continual proportionals; or when the straight line A C is so divided in B, that A C. A B. B C are in continual proportion. And if the same line A C be again divided in D, so as that A C. C D. A D be continual proportionals; then also A C. A B. A D will be continual proportionals; and in like manner, though in contrary order, C A. C D. C B will be continual proportionals; which cannot happen in any line otherwise divided.

A B C
----|----|----
D

25. If there be three continual proportionals, and again, three other continual proportions, which have the same middle term, their extremes will be in reciprocal proportion. For let A. B. C and D. B. E be continual proportionals, I say A. D :: E. C shall be proportionals. For the proportion of A to D is compounded of the proportions of A to B, and of B to D; and the proportion of E to C is compounded of those of E to B, that is, of B to D, and of B to C, that is, of A to B. Wherefore, by equality, A. D :: E. C are proportionals.

[Sidenote: Comparison of arithmetical and geometrical proportion.]

If any two unequal quantities be made extremes, and there be interposed betwixt them any number of means in geometrical proportion, and the same number of means in arithmetical proportion, the several means in geometrical proportion will be less than the several means in arithmetical proportion. For betwixt A the lesser, and E the greater extreme, let there be interposed three means, B, C, D, in geometrical proportion, and as many more, F, G, H, in arithmetical proportion; I say B will be less than F, C than G, and D than H. For first, the difference between A and F is the same with that between F and G, and with that between G and H, by the definition of arithmetical proportion; and therefore, the difference of the proportionals which stand next to one another, to the difference of the extremes, is, when there is but one mean, half their difference; when two, a third part of it; when three, a quarter, &c.; so that in this example it is a quarter. But the difference between D and E, by art. 17, is more than a quarter of the difference between the extremes, because the proportion is geometrical, and therefore the difference between A and D is less than three quarters of the same difference of the extremes. In like manner, if the difference between A and D be understood to be divided into three equal parts, it may be proved, that the difference between A and C is less than two quarters of the difference of the extremes A and E. And lastly, if the difference between A and C be divided into two equal parts, that the difference between A and B is less than a quarter of the difference of the extremes A and E.

A A
- -
B F
--- ---
C G
----- -----
D H
------- -------
E E
--------- ---------

From the consideration hereof, it is manifest, that B, that is A together with something else which is less than a fourth part of the difference of the extremes A and E, is less than F, that is, than the same A with something else which is equal to the said fourth part. Also, that C, that is A with something else which is less than two fourth parts of the said difference, is less than G, that is, than A together with the said two-fourths. And lastly, that D, which exceeds A by less than three-fourths of the said difference, is less than H, which exceeds the same A by three entire fourths of the said difference. And in the same manner it would be if there were four means, saving that instead of fourths of the difference of the extremes we are to take fifth parts; and so on.

27. LEMMA. If a quantity being given, first one quantity be both added to it and subtracted from it, and then another greater or less, the proportion of the remainder to the aggregate, is greater where the less quantity is added and substracted, than where the greater quantity is added and substracted. Let B be added to and substracted from the quantity A; so that A - B be the remainder, and A + B the aggregate; and again, let C, a greater quantity than B, be added to and substracted from the same A, so that A - C be the remainder and A + C the aggregate; I say A - B. A + B :: A - C. A + C will be an hyperlogism. For A - B. A :: A - C. A is an hyperlogism of a greater antecedent to the same consequent; and therefore A - B. A + B :: A - C. A + C is a much greater hyperlogism, being made of a greater antecedent to a less consequent.

28. If unequal parts be taken from two equal quantities, and betwixt the whole and the part of each there be interposed two means, one in geometrical, the other in arithmetical proportion; the difference betwixt the two means will be greatest, where the difference betwixt the whole and its part is greatest. For let A B and A B be two equal quantities, from which let two unequal parts be taken, namely, A E the less, and A F the greater; and betwixt A B and A E let A G be a mean in geometrical proportion, and A H a mean in arithmetical proportion. Also betwixt A B and A F let A I be a mean in geometrical proportion, and A K a mean in arithmetical proportion; I say H G is greater than K I.

A E G H B
---+---+---+----
----+---+---+---
A F I K B

For in the first place we have this A B. A G :: B G. G E, by
analogism article 18.

Then by composition we have this A B + A G. A B :: B G + G E
that is, B E. B G.

And by taking the halves of the ½A B + ½A G. A B :: ½B G + ½G E,
antecedents this third that is, B H. B G.

And by conversion a fourth A B. ½A B + ½A G :: B G. B H.

And by division this fifth ½A B - ½A G. ½A B + ½A G
:: H G. B H.

And by doubling the first
antecedent and the first
consequent A B - A G. A B + A G :: H G. B H.

Also by the same method may be
found out this analogism A B - A I. A B + AI :: K I. B K.

Now seeing the proportion of A B to A E is greater than that of A B to A F, the proportion of A B to A G, which is half the greater proportion, is greater than the proportion of A B to A I the half of the less proportion; and therefore A I is greater than A G. Wherefore the proportion of A B - A G to A B + A G, by the precedent lemma, will be greater than the proportion of A B - A I to A B + A I; and therefore also the proportion of H G to B H will be greater than that of K I to B K, and much greater than the proportion of K I to B H, which is greater than B K; for B H is the half of B E, as B K is the half of B F, which, by supposition, is less than B E. Wherefore H G is greater than K I; which was to be proved.

Coroll. It is manifest from hence, that if any quantity be supposed to be divided into equal parts infinite in number, the difference between the arithmetical and geometrical means will be infinitely little, that is, none at all. And upon this foundation, chiefly, the art of making those numbers, which are called Logarithms, seems to have been built.

29. If any number of quantities be propounded, whether they be unequal, or equal to one another; and there be another quantity, which multiplied by the number of the propounded quantities, is equal to them all; that other quantity is a mean in arithmetical proportion to all those propounded quantities.

-------

CHAP. XIV.

OF STRAIT AND CROOKED, ANGLE AND
FIGURE.

1. The definition and properties of a strait line.—2. The definition and
properties of a plane superficies.—3. Several sorts of crooked
lines.—4. The definition and properties of a circular line.—5. The
properties of a strait line taken in a plane.—6.. The definition of
tangent lines.—7. The definition of an angle, and the kinds
thereof.—8. In concentric circles, arches of the same angle are to
one another, as the whole circumferences are.—9. The quantity of an
angle, in what it consists. —10. The distinction of angles, simply
so called.—11. Of strait lines from the centre of a circle to a
tangent of the same. —12. The general definition of parallels, and
the properties of strait parallels.—13. The circumferences of
circles are to one another, as their diameters are.—14. In
triangles, strait lines parallel to the bases are to one another, as
the parts of the sides which they cut off from the vertex.—15. By
what fraction of a strait line the circumference of a circle is
made.—16. That an angle of contingence is quantity, but of a
different kind from that of an angle simply so called; and that it
can neither add nor take away any thing from the same.—17. That the
inclination of planes is angle simply so called.—18. A solid angle
what it is.—19. What is the nature of asymptotes.—20. Situation, by
what it is determined.—21. What is like situation; what is figure;
and what are like figures.

[Sidenote: The definition end properties of a strait line.]

1. Between two points given, the shortest line is that, whose extreme points cannot be drawn farther asunder without altering the quantity, that is, without altering the proportion of that line to any other line given. For the magnitude of a line is computed by the greatest distance which may be between its extreme points; so that any one line, whether it be extended or bowed, has always one and the same length, because it can have but one greatest distance between its extreme points.

And seeing the action, by which a strait line is made crooked, or contrarily a crooked line is made strait, is nothing but the bringing of its extreme points nearer to one another, or the setting of them further asunder, a _crooked line_ may rightly be defined to be _that, whose extreme points may be understood to be drawn further asunder_; and a _strait line_ to be _that, whose extreme points cannot be drawn further asunder; and comparatively, a more crooked, to be that line whose extreme points are nearer to one another than those of the other, supposing both the lines to be of equal length_. Now, howsoever a line be bowed, it makes always a _sinus_ or cavity, sometimes on one side, sometimes on another; so that the same crooked line may either have its whole cavity on one side only, or it may have it part on one side and part on the other side. Which being well understood, it will be easy to understand the following comparisons of strait and crooked lines.

First, if a strait and a crooked line have their extreme points common, the crooked line is longer than the strait line. For if the extreme points of the crooked line be drawn out to their greatest distance, it will be made a strait line, of which that, which was a strait line from the beginning, will be but a part; and therefore the strait line was shorter than the crooked line, which had the same extreme points. And for the same reason, if two crooked lines have their extreme points common, and both of them have all their cavity on one and the same side, the outermost of the two will be the longest line.

Secondly, a strait line and a perpetually crooked line cannot be coincident, no, not in the least part. For if they should, then not only some strait line would have its extreme points common with some crooked line, but also they would, by reason of their coincidence, be equal to one another; which, as I have newly shown, cannot be.

Thirdly, between two points given, there can be understood but one strait line; because there cannot be more than one least interval or length between the same points. For if there may be two, they will either be coincident, and so both of them will be one strait line; or if they be not coincident, then the application of one to the other by extension will make the extended line have its extreme points at greater distance than the other; and consequently, it was crooked from the beginning.

Fourthly, from this last it follows, that two strait lines cannot include a superficies. For if they have both their extreme points common, they are coincident; and if they have but one or neither of them common, then at one or both ends the extreme points will be disjoined, and include no superficies, but leave all open and undetermined.

Fifthly, every part of a strait line is a strait line. For seeing every part of a strait line is the least that can be drawn between its own extreme points, if all the parts should not constitute a strait line, they would altogether be longer than the whole line.

[Sidenote: The definition and properties of a plane superficies.]

2. A _plane_ or _a plane superficies, is that which is described by a strait line so moved, that all the several points thereof describe several strait lines_. A strait line, therefore, is necessarily all of it in the same plane which it describes. Also the strait lines, which are made by the points that describe a plane, are all of them in the same plane. Moreover, if any line whatsoever be moved in a plane, the lines, which are described by it, are all of them in the same plane.

All other superficies, which are not plane, are crooked, that is, are either concave or convex. And the same comparisons, which were made of strait and crooked lines, may also be made of plane and crooked superficies.

For, first, if a plane and crooked superficies be terminated with the same lines, the crooked superficies is greater than the plane superficies. For if the lines, of which the crooked superficies consists, be extended, they will be found to be longer than those of which the plane superficies consists, which cannot be extended, because they are strait.

Secondly, two superficies, whereof the one is plane, and the other continually crooked, cannot be coincident, no, not in the least part. For if they were coincident, they would be equal; nay, the same superficies would be both plane and crooked, which is impossible.

Thirdly, within the same terminating lines there can be no more than one plane superficies; because there can be but one least superficies within the same.

Fourthly, no number of plane superficies can include a solid, unless more than two of them end in a common vertex. For if two planes have both the same terminating lines, they are coincident, that is, they are but one superficies; and if their terminating lines be not the same, they leave one or more sides open.

Fifthly, every part of a plane superficies is a plane superficies. For seeing the whole plane superficies is the least of all those, that have the same terminating lines; and also every part of the same superficies is the least of all those, that are terminated with the same lines; if every part should not constitute a plane superficies, all the parts put together would not be equal to the whole.

[Sidenote: Several sorts of crooked lines.]

3. Of straitness, whether it be in lines or in superficies, there is but one kind; but of crookedness there are many kinds; for of crooked magnitudes, some are congruous, that is, are coincident when they are applied to one other; others are incongruous. Again, some are ὁμοιομερεῖς or uniform, that is, have their parts, howsoever taken, congruous to one another; others are ἀνομοιομερεῖς or of several forms. Moreover, of such as are crooked, some are continually crooked, others have parts which are not crooked.

[Sidenote: Definition and properties of a circular line.]

4. If a strait line be moved in a plane, in such manner, that while one end of it stands still, the whole line be carried round about till it come again into the same place from whence it was first moved, it will describe a plane superficies, which will be terminated every way by that crooked line, which is made by that end of the strait line which was carried round. Now this superficies is called a CIRCLE; and of this circle, the unmoved point is the _centre_; the crooked line which terminates it, the _perimeter_; and every part of that crooked line, a _circumference_ or _arch_; the strait line, which generated the _circle_, is the _semidiameter_ or _radius_; and any strait line, which passeth through the centre and is terminated on both sides in the circumference, is called the _diameter_. Moreover, every point of the radius, which describes the circle, describes in the same time its own perimeter, terminating its own circle, which is said to be _concentric_ to all the other circles, because this and all those have one common centre.

Wherefore in every circle, all strait lines from the centre to the circumference are equal. For they are all coincident with the radius which generates the circle.

Also the diameter divides both the perimeter and the circle itself into two equal parts. For if those two parts be applied to one another, and the semiperimeters be coincident, then, seeing they have one common diameter, they will be equal; and the semicircles will be equal also; for these also will be coincident. But if the semiperimeters be not coincident, then some one strait line, which passes through the centre, which centre is in the diameter, will be cut by them in two points. Wherefore, seeing all the strait lines from the centre to the circumference are equal, a part of the same strait line will be equal to the whole; which is impossible.

For the same reason the perimeter of a circle will be uniform, that is, any one part of it will be coincident with any other equal part of the same.

[Sidenote: The properties of a strait line taken in a plane.]

5. From hence may be collected this property of a strait line, namely, that it is all contained in that plane which contains both its extreme points. For seeing both its extreme points are in the plane, that strait line, which describes the plane, will pass through them both; and if one of them be made a centre, and at the distance between both a circumference be described, whose radius is the strait line which describes the plane, that circumference will pass through the other point. Wherefore between the two propounded points, there is one strait line, by the definition of a circle, contained wholly in the propounded plane; and therefore if another strait line might be drawn between the same points, and yet not be contained in the same plane, it would follow, that between two points two strait lines may be drawn; which has been demonstrated to be impossible.

It may also be collected, that if two planes cut one another, their common section will be a strait line. For the two extreme points of the intersection are in both the intersecting planes; and between those points a strait line may be drawn; but a strait line between any two points is in the same plane, in which the points are; and seeing these are in both the planes, the strait line which connects them will also be in both the same planes, and therefore it is the common section of both. And every other line, that can be drawn between those points, will be either coincident with that line, that is, it will be the same line; or it will not be coincident, and then it will be in neither, or but in one of those planes.

As a strait line may be understood to be moved round about whilst one end thereof remains fixed, as the centre; so in like manner it is easy to understand, that a plane may be circumduced about a strait line, whilst the strait line remains still in one and the same place, as the _axis_ of that motion. Now from hence it is manifest, that any three points are in some one plane. For as any two points, if they be connected by a strait line, are understood to be in the same plane in which the strait line is; so, if that plane be circumduced about the same strait line, it will in its revolution take in any third point, howsoever it be situate; and then the three points will be all in that plane; and consequently the three strait lines which connect those points, will also be in the same plane.

[Sidenote: Definition of tangent lines.]

6. Two lines are said to _touch_ one another, which being both drawn to one and the same point, will not cut one another, though they be produced, produced, I say, in the same manner in which they were generated. And therefore if two strait lines touch one another in any one point, they will be contiguous through their whole length. Also two lines continually crooked will do the same, if they be congruous and be applied to one another according to their congruity; otherwise, if they be incongruously applied, they will, as all other crooked lines, touch one another, where they touch, but in one point only. Which is manifest from this, that there can be no congruity between a strait line and a line that is continually crooked; for otherwise the same line might be both strait and crooked. Besides, when a strait line touches a crooked line, if the strait line be never so little moved about upon the point of contact, it will cut the crooked line; for seeing it touches it but in one point, if it incline any way, it will do more than touch it; that is, it will either be congruous to it, or it will cut it; but it cannot be congruous to it; and therefore it will cut it.

[Sidenote: The definition of an angle, and the kinds thereof.]

7. An angle, according to the most general acceptation of the word, may be thus defined; _when two lines, or many superficies, concur in one sole point, and diverge every where else, the quantity of that divergence is an_ ANGLE. And an angle is of two sorts; for, first, it may be made by the concurrence of lines, and then it is a _superficial angle_; or by the concurrence of superficies, and then it is called a _solid angle_.

Again, from the two ways by which two lines may diverge from one another, superficial angles are divided into two kinds. For two strait lines, which are applied to one another, and are contiguous in their whole length, may be separated or pulled open in such manner, that their concurrence in one point will still remain; and this separation or opening may be either by circular motion, the centre whereof is their point of concurrence, and the lines will still retain their straitness, the quantity of which separation or divergence is an _angle_ simply so called; or they may be separated by continual flexion or curvation in every imaginable point; and the quantity of this separation is that, which is called an _angle of contingence_.

Besides, of superficial angles simply so called, those, which are in a plane superficies, are plane; and those, which are not plane, are denominated from the superficies in which they are.

Lastly, those are _strait-lined angles_, which are made by strait lines; as those which are made by crooked lines are _crooked-lined_; and those which are made both of strait and crooked lines, are _mixed angles_.

[Sidenote: In concentric circles, arches of the same angle are to one
another, as the whole circumferences are.]

8. Two arches intercepted between two radii of concentric circles, have the same proportion to one another, which their whole perimeters have to one another. For let the point A (in the first figure) be the centre of the two circles B C D and E F G, in which the radii A E B and A F C intercept the arches B C and E F; I say the proportion of the arch B C to the arch E F is the same with that of the perimeter B C D to the perimeter E F G. For if the radius A F C be understood to be moved about the centre A with circular and uniform motion, that is, with equal swiftness everywhere, the point C will in a certain time describe the perimeter B C D, and in a part of that time the arch B C; and because the velocities are equal by which both the arch and the whole perimeter are described, the proportion of the magnitude of the perimeter B C D to the magnitude of the arch BC is determined by nothing but the difference of the times in which the perimeter and the arch are described. But both the perimeters are described in one and the same time, and both the arches in one and the same time; and therefore the proportions of the perimeter B C D to the arch B C, and of the perimeter E F G to the arch E F, are both determined by the same cause. Wherefore B C D. B C :: E F G. E F are proportionals (by the 6th art. of the last chapter), and by permutation B C D. E F G :: B C. E F will also be proportionals; which was to be demonstrated.

[Sidenote: The quantity of an angle, in what it consists.]

9. Nothing is contributed towards the quantity of an angle, neither by the length, nor by the equality, nor by the inequality of the lines which comprehend it. For the lines A B and A C comprehend the same angle which is comprehended by the lines A E and A F, or A B and A F. Nor is an angle either increased or diminished by the absolute quantity of the arch, which subtends the same; for both the greater arch B C and the lesser arch E F are subtended to the same angle. But the quantity of an angle is estimated by the quantity of the subtending arch compared with the quantity of the whole perimeter. And therefore the quantity of an angle simply so called may be thus defined: _the quantity of an angle is an arch or circumference of a circle, determined by its proportion to the whole perimeter_. So that when an arch is intercepted between two strait lines drawn from the centre, look how great a portion that arch is of the whole perimeter, so great is the angle. From whence it may be understood, that when the lines which contain an angle are strait lines, the quantity of that angle may be taken at any distance from the centre. But if one or both of the containing lines be crooked, then the quantity of the angle is to be taken in the least distance from the centre, or from their concurrence; for the least distance is to be considered as a strait line, seeing no crooked line can be imagined so little, but that there may be a less strait line. And although the least strait line cannot be given, because the least given line may still be divided, yet we may come to a part so small, as is not at all considerable; which we call a point. And this point may be understood to be in a strait line which touches a crooked line; for an angle is generated by separating, by circular motion, one strait line from another which touches it, as has been said above in the 7th article. Wherefore an angle, which two crooked lines make, is the same with that which is made by two strait lines which touch them.

[Sidenote: The distinction of angles, simply so called.]

10. From hence it follows, that _vertical angles_, such as are A B C, D B F in the second figure, are equal to one another. For if, from the two semiperimeters D A C, F D A, which are equal to one another, the common arch D A be taken away, the remaining arches A C, D F will be equal to one another.

Another distinction of angles is into _right_ and _oblique_. _A right angle is that, whose quantity is the fourth part of the perimeter._ And the lines, which make a right angle, are said to be _perpendicular_ to one another. Also, of oblique angles, that which is greater than a right, is called an _obtuse angle_; and that which is less, an _acute angle_. From whence it follows, that all the angles that can possibly be made at one and the same point, together taken, are equal to four right angles; because the quantities of them all put together make the whole perimeter. Also, that all the angles, which are made on one side of a strait line, from any one point taken in the same, are equal to two right angles; for if that point be made the centre, that strait line will be the diameter of a circle, by whose circumference the quantity of an angle is determined; and that diameter will divide the perimeter into two equal parts.

[Sidenote: Of strait lines from the centre of a circle to a tangent of
the same.]

11. If a tangent be made the diameter of a circle, whose centre is the point of contact, a strait line drawn from the centre of the former circle to the centre of the latter circle, will make two angles with the tangent, that is, with the diameter of the latter circle, equal to two right angles, by the last article. And because, by the 6th article, the tangent has on both sides equal inclination to the circle, each of them will be a right angle; as also the semidiameter will be perpendicular to the same tangent. Moreover, the semidiameter, inasmuch as it is the semidiameter, is the least strait line which can be drawn from the centre to the tangent; and every other strait line, that reaches the tangent, will pass out of the circle, and will therefore be greater than the semidiameter. In like manner, of all the strait lines, which may be drawn from the centre to the tangent, that is the greatest which makes the greatest angle with the perpendicular; which will be manifest, if about the same centre another circle be described, whose semidiameter is a strait line taken nearer to the perpendicular, and there be drawn a perpendicular, that is, a tangent, to the same.

From whence it is also manifest, that if two strait lines, which make equal angles on either side of the perpendicular, be produced to the tangent, they will be equal.

[Sidenote: The general definition of parallels; the properties of strait
parallels.]

12. There is in Euclid a definition of strait-lined parallels; but I do not find that parallels in general are anywhere defined; and therefore for an universal definition of them, I say that _any two lines whatsoever, strait or crooked, as also any two superficies, are_ PARALLEL; _when two equal strait lines, wheresoever they fall upon them, make always equal angles with each of them_.

From which definition it follows; first, that any two strait lines, not inclined opposite ways, falling upon two other strait lines, which are parallel, and intercepting equal parts in both of them, are themselves also equal and parallel. As if A B and C D (in the third figure), inclined both the same way, fall upon the parallels A C and B D, and A C and B D be equal, A B and C D will also be equal and parallel. For the perpendiculars B E and D F being drawn, the right angles E B D and F D H will be equal. Wherefore, seeing E F and B D are parallel, the angles E B A and F D C will be equal. Now if D C be not equal to B A, let any other strait line equal to B A be drawn from the point D; which, seeing it cannot fall upon the point C, let it fall upon G. Wherefore A G will be either greater or less than B D; and therefore the angles E B A and F D C are not equal, as was supposed. Wherefore A B and C D are equal; which is the first.

Again, because they make equal angles with the perpendiculars B E and D F; therefore the angle C D H will be equal to the angle A B D, and, by the definition of parallels, A B and C D will be parallel; which is the second.

_That plane, which is included both ways within parallel lines, is called a_ PARALLELOGRAM.

Coroll. I. From this last it follows, that the angles A B D and C D H are equal, that is, that a strait line, as B H, falling upon two parallels, as A B and C D, makes the internal angle A B D equal to the external and opposite angle C D H.

Coroll. II. And from hence again it follows, that a strait line falling upon two parallels, makes the alternate angles equal, that is, the angle A G F, in the fourth figure, equal to the angle G F D. For seeing G F D is equal to the external opposite angle E G B, it will be also equal to its vertical angle A G F, which is alternate to G F D.

Coroll. III. That the internal angles on the same side of the line F G are equal to two right angles. For the angles at F, namely, G F C and G F D, are equal to two right angles. But G F D is equal to its alternate angle A G F. Wherefore both the angles G F C and A G F, which are internal on the same side of the line F G, are equal to two right angles.

Coroll. IV. That the three angles of a strait-lined plain triangle are equal to two right angles; and any side being produced, the external angle will be equal to the two opposite internal angles. For if there be drawn by the vertex of the plain triangle A B C (fig. 5) a parallel to any of the sides, as to A B, the angles A and B will be equal to their alternate angles E and F, and the angle C is common. But, by the 10th article, the three angles E, C and F, are equal to two right angles; and therefore the three angles of the triangle are equal to the same; which is the first. Again, the two angles B and D are equal to two right angles, by the 10th article. Wherefore taking away B, there will remain the angles A and C, equal to the angle D; which is the second.

Coroll. V. If the angles A and B be equal, the sides A C and C B will also be equal, because A B and E F are parallel; and, on the contrary, if the sides A C and C B be equal, the angles A and B will also be equal. For if they be not equal, let the angles B and G be equal. Wherefore, seeing G B and E F are parallels, and the angles G and B equal, the sides G C and C B will also be equal; and because C B and A C are equal by supposition, C G and C A will also be equal; which cannot be, by the 11th article.

Coroll. VI. From hence it is manifest, that if two radii of a circle be connected by a strait line, the angles they make with that connecting line will be equal to one another; and if there be added that segment of the circle, which is subtended by the same line which connects the radii, then the angles, which those radii make with the circumference, will also be equal to one another. For a strait line, which subtends any arch, makes equal angles with the same; because, if the arch and the subtense be divided in the middle, the two halves of the segment will be congruous to one another, by reason of the uniformity both of the circumference of the circle, and of the strait line.

[Sidenote: The circumferences of circles are to one another as their
diameters are.]

13. Perimeters of circles are to one another, as their semidiameters are. For let there be any two circles, as, in the first figure, B C D the greater, and E F G the lesser, having their common centre at A; and let their semidiameters be A C and A E. I say, A C has the same proportion to A E, which the perimeter B C D has to the perimeter E F G. For the magnitude of the semidiameters A C and A E is determined by the distance of the points C and E from the centre A; and the same distances are acquired by the uniform motion of a point from A to C, in such manner, that in equal times the distances acquired be equal. But the perimeters B C D and E F G are also determined by the same distances of the points C and E from the centre A; and therefore the perimeters B C D and E F G, as well as the semidiameters A C and A E, have their magnitudes determined by the same cause, which cause makes, in equal times, equal spaces. Wherefore, by the 13th chapter and 6th article, the perimeters of circles and their semidiameters are proportionals; which was to be proved.

[Sidenote: In triangles strait lines parallel to the bases are to one
another, as the parts of the sides which they cut off from
the vertex.]

14. If two strait lines, which constitute an angle, be cut by strait-lined parallels, the intercepted parallels will be to one another, as the parts which they cut off from the vertex. Let the strait lines A B and A C, in the 6th figure, make an angle at A, and be cut by the two strait-lined parallels B C and D E, so that the parts cut off from the vertex in either of those lines, as in A B, may be A B and A D. I say, the parallels B C and D E are to one another, as the parts A B and A D. For let A B be divided into any number of equal parts, as into A F, F D, D B; and by the points F and D, let F G and D E be drawn parallel to the base B C, and cut A C in G and E; and again, by the points G and E, let other strait lines be drawn parallel to A B, and cut B C in H and I. If now the point A be understood to be moved uniformly over A B, and in the same time B be moved to C, and all the points F, D, and B be moved uniformly and with equal swiftness over F G, D E, and B C; then shall B pass over B H, equal to F G, in the same time that A passes over A F; and A F and F G will be to one another, as their velocities are; and when A is in F, D will be in K; when A is in D, D will be in E; and in what manner the point A passes by the points F, D, and B, in the same manner the point B will pass by the points H, I, and C; and the strait lines F G, D K, K E, B H, H I, and I C, are equal, by reason of their parallelism; and therefore, as the velocity in A B is to the velocity in B C, so is A D to D E; but as the velocity in A B is to the velocity in B C, so is A B to B C; that is to say, all the parallels will be severally to all the parts cut off from the vertex, as A F is to F G. Wherefore, A F. G F :: A D. D E :: A B. B C are proportionals.

The subtenses of equal angles in different circles, as the strait lines B C and F E (in fig. 1), are to one another as the arches which they subtend. For (by art. 8) the arches of equal angles are to one another as their perimeters are; and (by art. 13) the perimeters as their semidiameters; but the subtenses B C and F E are parallel to one another by reason of the equality of the angles which they make with the semidiameters; and therefore the same subtenses, by the last precedent article, will be proportional to the semidiameters, that is, to the perimeters, that is, to the arches which they subtend.

[Sidenote: By what fraction of a strait line the circumference of a
circle is made.]

15. If in a circle any number of equal subtenses be placed immediately after one another, and strait lines be drawn from the extreme point of the first subtense to the extreme points of all the rest, the first subtense being produced will make with the second subtense an external angle double to that, which is made by the same first subtense, and a tangent to the circle touching it in the extreme points thereof; and if a strait line which subtends two of those arches be produced, it will make an external angle with the third subtense, triple to the angle which is made by the tangent with the first subtense; and so continually. For with the radius A B (in fig. 7) let a circle be described, and in it let any number of equal subtenses, B C, C D, and D E, be placed; also let B D and B E be drawn; and by producing B C, B D and B E to any distance in G, H and I, let them make angles with the subtenses which succeed one another, namely, the external angles G C D, and H D E. Lastly, let the tangent K B be drawn, making with the first subtense the angle K B C. I say the angle G C D is double to the angle K B C, and the angle H D E triple to the same angle K B C. For if A C be drawn cutting B D in M, and from the point C there be drawn L C perpendicular to the same A C, then C L and M D will be parallel, by reason of the right angles at C and M; and therefore the alterne angles L C D and B D C will be equal: as also the angles B D C and C B D will be equal, because of the equality of the strait lines B C and C D. Wherefore the angle G C D is double to either of the angles C B D or C D B; and therefore also the angle G C D is double to the angle L C D, that is, to the angle K B C. Again, C D is parallel to B E, by reason of the equality of the angles C B E and D E B, and of the strait lines C B and D E; and therefore the angles G C D and G B E are equal; and consequently G B E, as also D E B is double to the angle K B C. But the external angle H D E is equal to the two internal D E B and D B E; and therefore the angle H D E is triple to the angle K B C, &c.; which was to be proved.

Coroll. I. From hence it is manifest, that the angles K B C and C B D, as also, that all the angles that are comprehended by two strait lines meeting in the circumference of a circle and insisting upon equal arches, are equal to one another.

Coroll. II. If the tangent B K be moved in the circumference with uniform motion about the centre B, it will in equal times cut off equal arches; and will pass over the whole perimeter in the same time in which itself describes a semiperimeter about the centre B.

Coroll. III. From hence also we may understand, what it is that determines the bending or curvation of a strait line into the circumference of a circle; namely, that it is fraction continually increasing in the same manner, as numbers, from one upwards, increase by the continual addition of unity. For the indefinite strait line K B being broken in B according to any angle, as that of K B C, and again in C according to a double angle, and in D according to an angle which is triple, and in E according to an angle which is quadruple to the first angle, and so continually, there will be described a figure which will indeed be rectilineal, if the broken parts be considered as having magnitude; but if they be understood to be the least that can be, that is, as so many points, then the figure described will not be rectilineal, but a circle, whose circumference will be the broken line.

Coroll. IV. From what has been said in this present article, it may also be demonstrated, that an angle in the centre is double to an angle in the circumference of the same circle, if the intercepted arches be equal. For seeing that strait line, by whose motion an angle is determined, passes over equal arches in equal times, as well from the centre as from the circumference; and while that, which is from the circumference, is passing over half its own perimeter, it passes in the same time over the whole perimeter of that which is from the centre, the arches, which it cuts off in the perimeter whose centre is A, will be double to those, which it makes in its own semiperimeter, whose centre is B. But in equal circles, as arches are to one another, so also are angles.

It may also be demonstrated, that the external angle made by a subtense produced and the next equal subtense is equal to an angle from the centre insisting upon the same arch; as in the last diagram, the angle G C D is equal to the angle C A D; for the external angle G C D is double to the angle C B D; and the angle C A D insisting upon the same arch C D is also double to the same angle C B D or K B C.

[Sidenote: That an angle of contingence is quantity, but of a different
kind from that of an angle simply so called; and that it can
neither add nor take away anything from the same.]

16. An angle of contingence, if it be compared with an angle simply so called, how little soever, has such proportion to it as a point has to a line; that is, no proportion at all, nor any quantity. For first, an angle of contingence is made by continual flexion; so that in the generation of it there is no circular motion at all, in which consists the nature of an angle simply so called; and therefore it cannot be compared with it according to quantity. Secondly, seeing the external angle made by a subtense produced and the next subtense is equal to an angle from the centre insisting upon the same arch, as in the last figure the angle G C D is equal to the angle C A D, the angle of contingence will be equal to that angle from the centre, which is made by A B and the same A B; for no part of a tangent can subtend any arch; but as the point of contact is to be taken for the subtense, so the angle of contingence is to be accounted for the external angle, and equal to that angle whose arch is the same point B.

Now, seeing an angle in general is defined to be the opening or divergence of two lines, which concur in one sole point; and seeing one opening is greater than another, it cannot be denied, but that by the very generation of it, an angle of contingence is quantity; for wheresoever there is greater and less, there is also quantity; but this quantity consists in greater and less flexion; for how much the greater a circle is, so much the nearer comes the circumference of it to the nature of a strait line; for the circumference of a circle being made by the curvation of a strait line, the less that strait line is, the greater is the curvation; and therefore, when one strait line is a tangent to many circles, the angle of contingence, which it makes with a less circle, is greater than that which it makes with a greater circle.

Nothing therefore is added to or taken from an angle simply so called, by the addition to it or taking from it of never so many angles of contingence. And as an angle of one sort can never be equal to an angle of the other sort, so they cannot be either greater or less than one another.

From whence it follows, that an angle of a segment, that is, the angle, which any strait line makes with any arch, is equal to the angle which is made by the same strait line, and another which touches the circle in the point of their concurrence; as in the last figure, the angle which is made between G B and B K is equal to that which is made between G B and the arch B C.

[Sidenote: That the inclination of planes is angle simply so called.]

17. An angle, which is made by two planes, is commonly called the inclination of those planes; and because planes have equal inclination in all their parts, instead of their inclination an angle is taken, which is made by two strait lines, one of which is in one, the other in the other of those planes, but both perpendicular to the common section.

[Sidenote: A solid angle what it is.]