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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.

THE

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A

A.'
MAGNITUDE is said to be contained once, in any'magni- Book V.

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.

III. Omitted.

B.
Multiplies, which contain their parts the same

number of times, are called equimultiples of
their parts : And the parts are called similar parts U
of their multiples.

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,
D are called fimilar parts of A, G.

IV.

part of A.

Book. V.

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,
when they are terminated, and of the same kind.

C.
In a ratio, the first named magnitude is called the Antecedent
Term, and the other the Confequent.

V.
The first of four magnitudes is said to have the same ratio to the

fecond which the third has to the fourth, when as many
times as any multiple of the first contains the second, so many
times does the fame multiple of the third contain 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.
If fome multiple of the first contain the second, a greater num-

ber of times than the same multiple of the third
contains the fourth, then the first is said to have M
to the second a greater ratio than the third 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.
Thus, if of A, C there can be taken such equi-
multiples MA, MC, that one of them MA con-

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.

D.
Magnitudes are said to be continual proportionals, when the

first has to the second the same ratio that the second has to
the third ; and the second to the third the same that the third
'has to the fourth; and so on.

E.
In three proportionals, the second is said to be a mean propor-
tional between the other two; and in

any
number of

proportionals, the first and the last of them are called the Extremes, and the others are called Means,

X

X.

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.

XI.
Of four 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.

F.
In 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
of the first to the second, and of the ratio of the second to
the third, and of the ratio of the third to the fourth, and so

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
to B, and of B to C, and of C to D, whether these ratios be
the fame with one another, or not : but if they be the same,
the ratio of A to C is also said to be duplicate of the ratio
of A to B, and the ratio of A to D triplicate of the ratio of

A to B.
In like manner, any ratio which is the same with that of A to

D, is said to be compounded of the ratios of A to B, B to C,
and C to D, or of any ratios which are the same with them.

XII
In proportionals, one antecedent is said to be homologous to

another antecedent, as also one consequent to another.
Changes in the order or magnitude of proportionals are made
various ways, some of which are the following.

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.

XVI.
By Division, 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. Prop. XVII. Book V.

XVII.
By Conversion, it is inferred, that the first is to its excess

above

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.
Thus, if A, B, C, D, be magnitudes in one rank, and E, F,

G, H, as many in another; then, if A be to A, B, C, D.
B as E to F, and B to C as F to G, and C
to D as G to H; it is inferred, by equality, E, F, G, H.
that A is to D as E to H. But if A be to B
as G to H, and B to C as F to G, and C to D as E to F,
it is inferred, by perturbate equality, that A is to D as E

to H.

A X I OM S.

J.
Equal magnitudes contain the same magnitude, the fame num-

ber of times ; and the same contains equals the fame number
of times,

II. That magnitude which contains the fame a greater number of times than another does, is greater than that other,

III.
That magnitude which is contained a greater number of times
than another, in the same magnitude, is less than that other.

IV.
Equimultiples of the fame, or of equal magnitudes, are equal

to one another,

V

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