Book IV. the third part of the whole, contains five; and the arch AB, which is the fifth part of the whole, contains three; therefore C 30. 3 BC, their difference, contains two of the fame parts: Bifect BC in E; therefore BE, EC are, each of them, the fifteenth part of the whole circumference ABCD: Therefore, if the ftraight lines BE, EC be drawn, and ftraight lines equal to d1. 4. them be placed around in the whole circle, an equilateral and equiangular quindecagon fhall be infcribed in it. Which was to be done.

And, in the fame manner as was done in the pentagon, if, through the points of divifion, made by inscribing the quinde. cagon, ftraight lines be drawn touching the circle, an equilateral and equiangular quindecagon fhall be defcribed about it; And likewife, as in the pentagon, a circle may be inscribed in a given equilateral and equiangular quindecagon, and circumfcribed about it.

THE

THE

ELEMENTS

OF

EU C L I D.

A

BOOK V.

DEFINITIONS.

A. '

MAGNITUDE is faid to be contained once, in any magni- Book V. tude not lefs than it, but lefs than its double: and it is faid to be contained twice, in any magnitude not lefs than its double, but lefs than its triple; and three times, in any not lefs than its triple, but lefs 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 lefs a certain number of times exactly, is faid to be a multiple of the less.

III.
B.

Omitted.

Multiplies, which contain their parts the fame number of times, are called equimultiples of their parts: And the parts are called fimilar parts of their multiples.

:

Thus, if A be exactly three times B; then A is faid to be a multiple of B and B is faid to be a part of A. 'If A be triple of B, and C alfo triple of D; then A, C are called equimultiples of B, D; and B, Dare called fimilar parts of A, C.

IV.

I

A B

C D

See N.

Magnitudes are faid to have a ratio to one another, when the lefs can be multiplied, fo as to exceed the other; that is, when they are terminated, and of the fame 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 ufually expreffed by faying, the firft is to the fecond as the third to the fourth.

VII.

M

If fome multiple of the first contain the fecond, a greater num-
ber of times than the fame multiple of the third
contains the fourth, then the first is faid to have
to the second a greater ratio than the third has
to the fourth; and, on the contrary, the third is
faid to have to the fourth a lefs ratio than the first
has to the fecond.

Thus, if of A, C there can be taken fuch equi

But

multiples MA, MC, that one of them MA con-
tains B oftener than the other MC contains D ;
then A has to B a greater ratio than C has to D;
and C has to D a lefs ratio than A has to B.
if no fuch equimultiples can be taken; that is, if
every multiple of A contains B as many times as
the fame multiple of C contains D, then A is to B
as C to D.
VIII. & IX. Omitted.
D.

A B

C D

M

I

Magnitudes are faid to be continual proportionals, when the first has to the second the fame ratio that the fecond has to the third; and the fecond to the third the fame that the third has to the fourth; and so on.

E.

In three proportionals, the second is said to be a mean proportional between the other two; and in any number of proportionals, the firft and the laft of them are called the Extremes, and the others are called Means,

X.

X.

The first of three proportionals is faid to have to the third the duplicate ratio of that which it has to the fecond.

XI.

Of four continual proportionals, the firft is faid to have to the fourth the triplicate ratio of that which it has to the second; and fo on, quadruplicate, &c. increasing the denomination ftill by unity.

F.

In any number of magnitudes of the fame kind, the first is faid to have to the laft of them the ratio compounded of the ratio of the first to the fecond, and of the ratio of the fecond to the third, and of the ratio of the third to the fourth, and fo on to the laft.

Thus, if A, B, C, D be magnitudes of the fame kind, the ratio of A to D is faid 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 fame with that of A to D, is faid to be compounded of the ratios of A to B, B to C, and C to D, or of any ratios which are the fame with them.

XII

In proportionals, one antecedent is faid to be homologous te
another antecedent, as alfo one confequent to another.
Changes in the order or magnitude of proportionals are made
various ways, fome of which are the following.

XIII.

By Alternation. When there are four proportionals; it is inferred, by alternation, that the first is to the third as the fe cond to the fourth; as is fhewn in Prop. XVI. Book V. XIV.

By Inverfion, it is inferred, that the second is to the first as the fourth to the third. Prop. B. Book V.

XV.

By Compofition, it is inferred, that the first, together with the fecond, is to the second, as the third, together with the fourth, is to the fourth. Prop. XVIII, Book V.

By Divifion, it is inferred, fecond, is to the second fourth is to the fourth.

XVI.

that the excess of the first above the
as the excess of the third above the
Prop. XVII. Book V.
XVII.

By Converfion, it is inferred, that the firft is to its excess

above

BOOK V.

Book V.

above the fecond, as the third to its excefs above the fourth, Prop. E. Book V.

XVIII. & XIX.

By Equality. When there are two ranks, each of them containing the fame 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 firft to the fecond of the other rank; and the fecond to the third, as the fecond to the third; and fo on; then it is inferred, by equality, that the first is to the laft of the first rank as the firfl 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 laft 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 laft of the first rank, as the first to the laft of the other rank. Prop. XXIII. Book V.

F.

Thus, if A, B, C, D, be magnitudes in one rank, and E,
G, H, as many in another; then, if A be to A, B, C, D.
Bas E to F, and B to C as F to G, and C

to Das 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.

AXIOM S.
J.

Equal magnitudes contain the fame magnitude, the fame number of times; and the fame 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 fame magnitude, is lefs than that other.

IV.

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

to one another,