the same parts, and BE, EC are, each of them, the fifteenth part of the whole circumference A B CD, and the figure A BECDF is equilateral. Because each of its angles stands upon thirteen-fifteenths of the circumference, it is also equiangular (III. 27). Therefore a regular quindecagon is inscribed in the circle AB C. Q. E. F.

In the same manner as was done in the pentagon, if through the angular points of the inscribed quindecagon, straight lines be drawn touching the circle, a regular quindecagon will be described about it; and likewise, as in the case of the pentagon, a circle may be inscribed in a given regular quindecagon, and circumscribed about it.

Otherwise.-Find two sides of the regular pentagon, and one side of the regular trigon inscribed in the circle, and commencing at the same point in the circumference. Join the points in which the second side of the pentagon and the side of the trigon terminate. The chord thus drawn will be the side of the inscribed pentedecagon or quindecagon. Because of such equal parts as the whole circumference contains fifteen, the arc subtended by the two sides of the regular pentagon contains two-fifths or six, and the arc subtended by the side of the trigon contains one-third or five. Therefore their difference contains one. fifteenth or one of those parts, which is the side of the quindecagon required.

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.

The term part is here evidently understood in a restricted sense, and one which is commonly expressed by the phrase aliquot part. Better terms are measure or submultiple, either of which signifies the same as part or aliquot part, and conveys a more distinct notion of the meaning.

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.

The meaning of this definition, taken in connexion with the preceding one, will be best understood by adopting two distinct terms which are correlative; viz., multiple, and submultiple. Thus, if one magnitude contains another an exact number of times, the greater magnitude is called the multiple of the smaller; and the smaller, the submultiple of the greater.

The mutual relation of two magnitudes of the same kind to one another, in respect of quantity, is called their, ratio.

The term ratio is employed to express the relation of two like magnitudes to each other, whether they be commensurable or incommensurable, that is, whether they have a common measure or not. Thus, the diagonal of a square has a certain ratio to the side of the square; but this ratio cannot be expressed, like many others, in commensurate terms; for their common measure, or common unit is unknown.

IV

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

This definition is intended as a test of the likeness or similarity of any two magnitudes; for unless the one can be multiplied so as to exceed the other in magnitude, they cannot be said to be of the same kind, and so cannot have any ratio to each 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

H

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.

This is the most important definition of the Fifth Book. Upon it, as a hinge or centre, turns the whole doctrine of proportion delivered in this Book, and applied in the sixth and subsequent Books. Volumes have been written to explain its meaning, and yet after all it is very simple. It is plain in the first place, that of any four magnitudes such as are spoken of in the definition, the first two must be homogeneous, or both of the same kind, and the last two must be homogeneous, or both of the same kind; but these two kinds may be different in themselves; that is, each pair may be heterogeneous, or of a different kind. Secondly, it is plain that by equimultiples of two magnitudes, is meant that each magnitude is taken or repeated the same number of times. Now, by taking equimultiples of the first and third of the supposed magnitudes, and equimultiples of the second and fourth of the same magnitudes, if it can be shown from the nature of the case to which this test is applied, that when the multiple of the first is greater than that of the second, the multiple of the third must also be greater than that of the fourth; or that when the multiple of the first is equal to that of the second, the multiple of the third must be equal to that of the fourth; or, lastly, when the multiple of the first is less than that of the second, the multiple of the third must be less than that of the fourth; then, it necessarily follows according to this definition, that the first has to the second the same ratio that the third has to the fourth; that is, that there is an equality of ratios between the first pair and the second pair of magnitudes. The application of this definition to particular cases, however, will be sure to render it much more clear and distinct to the learner.

VI.

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

This definition merely explains the term proportional as applied to magnitudes such as are supposed in the preceding definition, which constitutes the test of proportionality.

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 econd, 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.

This definition becomes plain and easy after the fifth definition is understood. It may be be considered as a test of non-proportionality,

VIII.

Analogy, or proportion, is the similitude of ratios.

This definition has been greatly objected to. Much of the objection may be removed by adopting the word sameness or equality, instead of similitude, this change being justified by the phraseology of the fifth definition itself.

IX.

Proportion consists in three terms at least.

When a proportion consists of three terms their order is continual, or such that the first has the same ratio to the second which the second has to the third.

X.

When three magnitudes are continual proportionals, the first is said to have to the third, the duplicate ratio of that which it has to the second. When three magnitudes are continual proportionals, the ratio of the first to the third is compounded of two equal ratios.-viz., the ratio of the first to the second, and the ratio of the second to the third; hence, it is called duplicate ratio.

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.

When four magnitudes are continual proportionals, the ratio of the first to the fourth is compounded of three equal ratios,-viz., the ratio of the first to the second, the ratio of the second to the third, and the ratio of the third to the fourth; hence, it is called triplicate ratio. In like manner, quadruplicate ratio is a ratio compounded of four equal ratios, &c.

A.

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 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 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 if A has to B the same ratio which E has to F; and B 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 L.

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 B to F, G to H, and K to L.

This definition marked A, is usually called the definition of compound ratio. It was supplied by Dr. Simson, and was considered by him to have originally belonged to the Elements, though not in the Greek text.

XII.

In proportionals, the antecedent terms are called homologous to one another, as also the consequents to one another.

Proportionals consist of a series of ratios. In any ratio, which, of course, consists of two terms or magnitudes, the first term of the ratio is called the antecedent, and the second term the consequent. In an ordinary proportion consisting of four terms. the first and the third, being the antecedents of the two ratios, are called homologous terms,-that is, terms which agree with one another as to their name; and the second and the fourth being the consequents of the two ratios, are also called homologous terms.

"Geometers make use of the following technical words or phrases to

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

[The memory need not be burdened with these explanations until the propositions be studied to which they refer.]

XIII.

Permutando, or alternando, by permutation or alternately. This phrase 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 shown in Prop XVI. of this Fifth 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 V.

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.-Prop. 18, Book V.

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.-Prop 17, Book V.

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 V.

XVIII.

Ex æquali (sc. distantia), or ex æquo, from equality of distance: when there is any number of magnitudes more than two, and as many others, such 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."

ΧΙΧ.

Ex æquali, 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 Prop. 22, Book V.

XX.

Ex æquali in proportione perturbatâ seu inordinatâ, from equality in