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Book IV. the third part of the whole, contains five ; and the arch AB,
W which is the fifth part of the whole, contains three; therefore C 30. 3. BC, their difference, contains two of the fame parts : Biseet
BC in E; therefore BE, EC are, each of them, the fifteenth part of the whole circumference ABCD: Therefore, if the
straight lines BE, EC be drawn, and straight lines equal to d 1. 4. them be placed d around in the whole circle, an equilateral and
equiangular quindecagon shall be inscribed in it. " Which was to be done.
And, in the fame manner as was done in the pentagon, if, through the points of division, made by inscribing the quinde. cagon, 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 cquilateral and equiangular quindecagon, and circumscribed about it.
tude not less than it, but less than its double: and it is said to be contained twice, in any magnitude not less than its See N. double, but less than its triple; and three times, in any not less than its triple, but less than its quadruple: and so on.
1. A part of a magnitude, is that which is contained in the magnitude a certain number of times exactly.
II. A greater magnitude, which contains a less a certain number of times exactly, is said to be a multiple of the less.
number of times, are called equimultiples of
A B Thus, if Å be exactly three times B; then A is said to be a multiple of B: and B is said to be a C D
I 'If A be triple of B, and C also triple of D; then
A, C are called equimultiples of B, D; and B,
part of A.
IV. W Magnitudes are said to have a ratio to one another, when the
less can be multiplied, so as to exceed the other; that is,
fecond which the third has to the fourth, when as many
expressed by saying, the first is to the second as the third to
ber of times than the same multiple of the third
has to the second.
A B tains B oftener than the other MC contains D;
C D then A has to B a greater ratio than C has to D; and C has to D a less ratio than A has to B. But if no such equimultiples can be taken ; that is, if
i every multiple of A contains B as many times as 1 the same multiple of C contains D, then A is to B M as C to D.
VIII. & IX. Omitted.
first has to the second the same ratio that the second has to
proportionals, the first and the last of them are called the Extremes, and the others are called Means,
BOOK V. The firft of three proportionals is said to have to the third the duplicate ratio of that which it has to the second.
fourth the triplicate ratio of that which it has to the second ;
to have to the last of them the ratio compounded of the ratio
on to the last. Thus, if A, B, C, D be magnitudes of the same kind, the
ratio of A to D is said to be compounded of the ratios of A
A to B.
D, is said to be compounded of the ratios of A to B, B to C,
another antecedent, as also one consequent to another.
XIII. By Alternation. When there are four proportionals ; it is in. ferred, by alternation, that the first is to the third as the se. cond to the fourth ; as is shewn in Prop. XVI. Book V.
XIV. By Inversion, it is inferred, that the second is to the first as the fourth to the third. Prop. B. Book V.
XV. By Composition, 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. Prop. XVIII, Book v.
Book V. above the second, as the third to its excess above the fourth, Prop. E. Book V.
XVIII. & XIX. By Equality. When there are two ranks, each of them containing the same number of magnitudes more than two, and these magnitudes are proportionals, when taken two and two in a direct order in each rank; that is, the first to the second of the first rank, as the first to the second of the other rank ; and the second to the third, as the second to the third ; and so on; then it is inferred, by equality, that the first is to the last of the first rank as the first of the other rank to the last.. Prop. XXII. Book V.
XX By Perturbate Equality. When there are two ranks as before, and the magnitudes are proportionals when taken two and two in each rank, one in a direct, and the other in an inverse order; that is, the first to the second of the first rank, as the last but one to the last of the other rank; and the second to the third, as the last but two to the last but one ; and so on ; then it is inferred, by perturbate equality, that the first is to the last of the first rank, as the first to the last of the other
rank. Prop. XXIII. Book V.
G, H, as many in another; then, if A be to A, B, C, D.
A X I OM S.
ber of times ; and the same contains equals the fame number
II. That magnitude which contains the fame a greater number of times than another does, is greater than that other,
to one another,