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third part of the whole, contains five; and the circumference a b, which is the fifth part of the whole, contains three; therefore bc their difference contains two of the same parts. Bisect

a (iii. 30) bc in e; therefore be, ec are, each of them, the fifteenth part of the whole circumference a bcd. Therefore, if the straight lines be, e c be drawn, and straight lines equal to them be placed round (i. 4) in the whole circle, an equilateral and equiangular quindecagon shall be inscribed in it. Which was to be done.

And in the same manner as was done in the pentagon, if, through the points of

d division made by inscribing the quindecagon, straight lines be drawn touching the circle, an equilateral and equiangular quindecagon shall be described about it. And, likewise, as in the pentagon, a circle may be inscribed in a given equilateral and equiangular quindecagon, and circumscribed about it.

EXERCISES ON BOOK IV.

SECT. I.-PROBLEMS.

1. Given a line and two points, to describe a circle that shall pass through the points and touch the line.

2. Given a circle, to inscribe in it three equal circles which shall touch it and each other.

3. Given a circle, to inscribe in it four equal circles which shall touch it and each other.

4. Given a circle, to describe six other circles, each of which shall be equal to the given circle and be in contact with it and each other.

5. Given a quadrant of a circle, to inscribe a circle in it. 6. Given two circles, to draw a line that shall be a tangent to both.

7. Given a finite straight line, to describe on it an equilateral and equiangular decagon.

8. Given a line and a circle, to draw a tangent to the circle parallel to the line.

SECT. II.-THEOREMS. 9. If the three angles of a triangle be bisected, the bisecting lines shall meet in the same point.

10. If a circle be inscribed in a triangle, the rectangle contained by the radius of the circle and the sum of the three sides of the triangles is double of the triangle.

11. If an equilateral triangle be described about a circle, the straight lines which join the points of contact of the circle and the triangle contain another equilateral triangle, each of which is equal to one half of a side of the other triangle.

12. If a square be inscribed in a circle, it shall be double the square of the radius of the circle.

BOOK V.

DEFINITIONS.

I. A LESS magnitude is said to be a part of a greater magnitude, when the less measures the greater ; that is, when the less is contained a certain number of times exactly in the greater.

II. A greater magnitude is said to be a multiple of a less, when the greater is measured by the less ; that is, when the greater contains the less a certain number of times exactly.

III. Ratio is a mutual relation of two magnitudes of the same kind to one another, in respect of quantity.

IV. Magnitudes are said to have a ratio to one another, when the less can be multiplied so as to exceed the other.

V. The first of four magnitudes is said to have the same ratio to the second which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth ; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth ; or, if the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth ; or, if the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth.

VI. Magnitudes which have the same ratio are called proportionals. N. B. When four magnitudes are proportionals, it is usually expressed by saying, the first is to the second, as the third to the fourth.

VII. When of the equimultiples of four magnitudes (taken as in the fifth definition), the multiple of the first is greater than that of the second, but the multiple of the third is not greater than the multiple of the fourth; then the first is said to have to the second a greater ratio than the third magnitude has to the fourth ; and, on the contrary, the third is said to have to the fourth a less ratio than the first has to the second.

VIII. Analogy, or proportion, is the similitude of ratios.
IX. Proportion consists in three terms at least.

X. When three magnitudes are proportionals, the first is said to have to the third the duplicate ratio of that which it has to the second.

XI. When four magnitudes are continual proportionals, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on, quadruplicate, &c. increasing the denomination still by unity, in any number of proportionals.

Definition a, to wit, of compound ratio. When there are any number of magnitudes of the same kind, the first is said to have to the last of them the ratio compounded of the ratio which the first has to the second, and of the ratio which the second has to the

a

third, and of the ratio which the third has to the fourth, and so on unto the last magnitude.

For example, if a, b, c, d be four magnitudes of the same kind, the first a is said to have to the last d the ratio compounded of the ratio of a to b, and the ratio of b to c, and the ratio of c to d; or, the ratio of to d is said to be compounded of the ratios of a to b, b to c, and c to d: And if a has to b the same ratio which e has to f; and

to c the same ratio that

g has to h; and c to d the same that k has to l; then by this definition, a is said to have to d the ratio compounded of ratios which are the same with the ratios of e to f, g to h, and k to l. And the same thing is to be understood when it is more briefly expressed by saying, a has to d the ratio compounded of the ratios of e to f, g to h, and k to 1.

In like manner, the same things being supposed, if m has to n the same ratio which a has to d ; then, for shortness sake, m is said to have to n the ratio compounded of the ratios of e to f, g to h, and k to 1.

XII. In proportionals, the antecedent terms are called homologous to one another, as also the consequents to one another. Geometers make use of the

following technical words to signify certain ways of changing either the order or magnitude of proportionals, so as that they continue still to be proportionals.

XIII. Permutando, or alternando, by permutation, or alternately. This word is used when there are four proportionals, and it is inferred that the first has the same ratio to the third which the second has to the fourth ; or that the first is to the third as the second to the fourth : as is shewn in the 16th Prop. of this 5th Book.

XIV. Invertendo, by inversion ; when there are four proportionals, and it is inferred that the second is to the first as the fourth to the third. Prop. B. Book 5.

XV. Componendo, by composition ; when there are four proportionals, and it is inferred that the first, together with the second, is to the second as the third, together with the fourth, is to the fourth. 18th Prop. Book 5.

XVI. Dividendo, by division; when there are four proportionals, and it is inferred that the excess of the first above the second is to the second as the excess of the third above the fourth is to the fourth. 17th Prop. Book 5.

XVII. Convertendo, by conversion ; when there are four proportionals, and it is inferred that the first is to its excess above the second as the third to its excess above the fourth. Prop. E. Book 5.

XVIII. Ex aequali (sc. distantia), or ex æquo, from equality of distance ; when there is any number of magnitudes more than two, and as many others, so that they are proportionals when taken two and two of each rank, and it is inferred that the first is to the last of the first rank of magnitudes as the first is to the last of the others. Of this there are the two following kinds, which arise from the different order in which the magnitudes are taken, two and two.

XIX. Ex equali, from equality. This term is used simply by itself, when the first magnitude is to the second of the first rank as the first to the second of the other rank ; and as the second is to the third of the first rank, so is the second to the third of the other; and so on in order. And the inference is as mentioned in the preceding definition, whence this is called." ordinate proportion.” It is demonstrated in 22d Prop. Book 5.

XX. Ex cequali in proportione perturbata seu inordinata, from equality in perturbate or disorderly proportion.* This term is used when the first magnitude is to the second of the first rank as the last but one is to the last of the second rank; and as the second is to the third of the first rank, so is the last but two to the last but one of the second rank ; and as the third is to the fourth of the first rank, so is the third from the last to the last but two of the second rank, and so on in a cross order : and the inference is as in the 18th definition. It is demonstrated in the 23d Prop. of Book 5.

AXIOMS. I. Equimultiples of the same or of equal magnitudes are equal to one another.

II. Those magnitudes of which the same or equal magnitudes are equimultiples are equal to one another.

III. A multiple of a greater magnitude is greater than the same multiple of a less.

IV. That magnitude, of which a multiple is greater than the same multiple of another, is greater than that other magnitude.

PROPOSITION I.—THEOREM. If any number of magnitudes be equimultiples of as many others, each of

each ; what multiple soever any one of them is of its part, the same multiple

shall all the first magnitudes be of all the others. LET any number of magnitudes a b, cd be equimultiples of as many others e, f, each of each : whatsoever multiple a b is of e, the same multiple shall a b and cd together be of e and fto- al gether.

Because a b is the same multiple of e that cd is of f, as many magnitudes as are in a b equal to e, so many are there 81 in cd equal to f. Divide a b into magnitudes equal to e, viz. ag, gb; and cd into ch, hd equal each of them to f:

b the number therefore of the magnitudes ch, hd, shall be equal to the number of the others ag, gb; and because ag is equal to e, and ch to f, therefore a g and ch together are equal to (i. 2 ax.) e and f together. For the same reason, because g b is equal to e. and h d to f; gb and h d together are equal to e and f together. Wherefore as many magni- h tudes as are in a b equal to e, so many are there in a b, cd together equal to e and f together. Therefore, whatsoever multiple a ħ is of e, the same multiple is a b and cd together d of e and f together.

Therefore, if any magnitudes, how many soever, he equimultiples of as many, each of each, whatsoever multiple any one of them is of its part, the same multiple shall all the first magnitudes be of all the other : for the same demonstration holds in any number of magnitudes, which was here applied to two. Q. E. D.

* Archimedes de Sphæra et Cylindro, Prop. 4. lib. 2.

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