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Book IV.

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

THE

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Lefs magnitude is faid to be a part of a greater magnitude, when the lefs measures the greater, that is, when the lefs is contained a certain number of times exactly in the greater.'

II.

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A greater magnitude is faid to be a multiple of a lefs, when the greater is measured by the lefs, that is, when the greater contains the lefs a certain number of times exactly.'

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Ratio is a mutual relation of two magnitudes of the fame See N • kind to one another, in refpect of quantity.'

IV.

Magnitudes are said to have a ratio to one another, when the lefs can be multiplied fo as to exceed the other.

V.

The first of four magnitudes is faid to have the fame ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the firft and third being taken, and any equimultiples whatsoever of the fecond and fourth; if the multiple of the first be less than that of the fecond, the multiple of the third is alfo lefs than that of the fourth; or, if the multiple of the first be equal to that of the fecond, the multiple of the third is alfo equal to that of the fourth H 4

h;

at,

Book V.

See N.

or, if the multiple of the first be greater than that of the
fecond, the multiple of the third is also greater than that of
the fourth.
VI.

Magnitudes which have the fame ratio are called proportionals,
N. B. When four magnitudes are proportionals, it is ufu-
ally expreffed by faying, the first is to the fecond, as the
third to the fourth."

VII.

When of the equimultiples of four magnitudes (taken as in the fifth definition) the multiple of the firft is greater than that of the fecond, but the multiple of the third is not greater than the multiple of the fourth; then the firft is faid to have to the fecond a greater ratio than the third magnitude 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.

VIII.

"Analogy, or proportion, is the fimilitude of ratios."

IX.

Proportion confifts in three terms at least.

X.

When three magnitudes are proportionals, the firft is faid to have to the third the duplicate ratio of that which it has to the fecond.

XI.

When four magnitudes are continual proportionals, the firft is faid to have to the fourth the triplicate ratio of that which it has to the fecond, and fo on, quadruplicate, &c. increasing the denomination ftill by unity, in any number of proportionals.

Definition A, to wit, of compound ratio. When there are any number of magnitudes of the fame kind, the firft is faid to have to the laft of them the ratio compounded of the ratio which the firft has to the second, and of the ratio which the second has to the third, and of the ratio which the third has to the fourth, and fo on unto the last magnitude.

For example, If A, B, C, D be four magnitudes of the fame kind, the firft A is faid to have to the laft D the ratio compounded of the ratio of A to B, and of the ratio of B to C, and of the ratio of C to D; or, the ratio of A to D is said to be compounded of the ratios of A to B, B to C, and C to D: And

E

And if A has to B, the fame ratio which E has to F; and B Book V.
to C, the fame ratio that G has to H; and C to D, the fame
that K has to L; then, by this definition, A is faid to have
to 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 expreffed, by
faying A has to D the ratio compounded of the ratios of E
to F, G to H, and K to L.

In like manner, the fame things being supposed, if M has to
N the fame ratio which A has to D; then, for fhortnefs fake,
Mis faid to have to N, the ratio compounded of the ratios of
E to F, G to H, and K to L.

XII.

In proportionals, the antecedent terms are called homologous to
one another, as also the confequents to one another.
'Geometers make ufe of the following technical words to fig-
⚫nify certain ways of changing either the order or magni-
'tude of proportionals, fo as that they continue still to be
'proportionals,'

XIII.

Permutando, or alternando, by permutation, or alternately; See N. this word is ufed when there are four proportionals, and it is inferred, that the first has the same ratio to the third, which the fecond has to the fourth; or that the first is to the third, as the second to the fourth: As is fhewn in the 16th prop. of this 5th book.

XIV.

Invertendo, by inverfion: When there are four proportionals, and it is inferred, that the fecond is to the firft, as the fourth to the third. Prop. B. book 5.

XV.

Componendo, by compofition; when there are four proportionals, and it is inferred, that the firft, together with the fe cond, is to the fecond, as the third, together with the fourth, is to the fourth. 18th prop. book 5.

XVI.

Dividendo, by divifion; when there are four proportionals, and it is inferred, that the excefs of the firit above the fecond, is to the fecond, as the excefs of the third above the fourth, is to the fourth. 17th prop. book 5.

XVII.

Convertendo, by converfion; when there are four proportion

als,

Book V. als, and it is inferred, that the first is to its excefs above the fecond, as the third to its excefs above the fourth. Prop. E. book 5.

XVIII.

Ex aequali (fc. diftantia), or ex aequo, from equality of distance; when there is any number of magnitudes more than two, and as many others, fo that they are proportionals when taken two and two of each rank, and it is inferred, that the first is to the laft 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 arife from the different order in which the magnitudes are taken two and two.'

XIX.

Ex aequali, from equality; this term is ufed fimply by itself, when the firft magnitude is to the fecond of the firft rank, as the first to the second of the other rank; and as the fecond is to the third of the first rank, fo is the fecond to the third of the other; and fo on in order, and the inference is as mentioned in the preceeding definition; whence this is called ordinate proportion. It is demonftrated in 22d prop. book 5.

XX.

Ex aequali, in proportione perturbata, feu inordinata; from equality, in perturbate or diforderly proportion +; this term is ufed when the first magnitude is to the fecond 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, fo is the laft but two to the last but one of the second rank; and as the third is to the fourth of the first rank, fo is the third from the last to the last but two of the fecond rank; and fo on in a crofs order: And the inference is as in the 18th definition. It is demonftrated in the 23. prop. of book 5.

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QUIMULTIPLES of the fame, or of equal magnitudes, are equal to one another.

↑ 4. Prop. lib. 2. Archimedis de fphaera et cylindro.

II. Thofe

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