if the multiple of the first be greater than that of the second, the mul. tiple 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 second, 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 p'ain and easy after the fifth definition is understood. It may 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 sume ess 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.

Δ.

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's 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 18 a ratio compounded of four equal ratios, &o.

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 defini. tion, 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.

It was

This definition marked A, is usually called the definition of compound ratio. 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 propositione be studied to which they refer.]

XIII.

This

Permutando, or alternando, by permutation or alternately. 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."

XIX.

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 perturbatá seu inordinatâ from equality in

perturbate or disorderly proportion (Prop. 4, Lib. II. Archimedis de sphæra et cylindro). The term is used when the first magnitude is to the second 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, so is the last but two to the last but one of the second rank; and as the third is to the fourth of the first rank, so is the third from the last to the last but two of the second rank; and so on in a cross order: and the inference is as in the 18th definition. It is demonstrated in Prop. 23, Book V.

[The Latin terms explained in these eight definitions are usually replaced by the English words which express their meaning. But they are retained in most editions of Euclid, as being useful in reading good old authors on geometry.}

AXIOM S.
I.

Equimultiples of the same, or of equal magnitudes, are equal to one

another.

II.

Those magnitudes, of which the same or equal magnitudes are equimultiples, are equal to one another.

This axiom means that equi-submuitiples of the same or of equal magnitudes, are equal.

III.

A multiple of a greater magnitude is greater than the same multiple of a less.

IV.

Of two magnitudes, that one of which a multiple is greater than the same multiple of the other, is the greater.

V.

A part or submultiple of a greater magnitude is greater than the same part or submultiple of a less magnitude.

VI.

Of two magnitudes, that one of which a part or submultiple is greater than the same part or submultiple of the other, is the greater of the two.

These two axioms are not Euclid's, but they are added as useful for reference. The magnitudes in the Fifth Book are usually represented by straight lines, for the sake of simplicity; but any other kind of figures may be employed to indicate magnitudes in general.

PROP. I. THEOREM.

If any number of magnitudes be equimultiples of as many other maynitudes, each of each; what multiple soever cny one of the first magnatudes is of its part, the same multiple is all the first magnitudes of all the other magnitudes.

Let any number of magnitudes AB and CD be equimultiples of as many others E and F, each of each. Whatsoever multiple A B is of E, the same multiple is A B and CD together, of E and F together. Divide AB into magnitudes each equal

to E, viz. A G and GB; and CD into CH and HD, each equal to F. Because the number of the magnitudes C H and HD is equal to the number of the others AG

A G в с и

E

F

D

and GB (Hyr.); and AG is equal to E, and CH to F(Const.). There fore AG and CH together are equal (I. Ax. 2) to E and F together. Because GB is equal to E, and HD to F. Therefore GB and HD together are equal to E and F together. Wherefore as many magnitudes as A B contains each equal to E, so many do A B and CD together contain each equal to E and F together. Therefore, whatsoever multiple AB is of E, the same multiple is AB and CD together, of E and F together. The same demonstration holds in any number of magnitudes, which is here applied to two. Therefore, if any number of magnitudes, be equimultiples of as many others, each of each; whatsoever multiple any one of them is of its part, the same multiple is all the first magnitudes of all the others. Q. E. D.

If the first magnitude be the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth; then the first together with the fifth is the same multiple of the second, that the third together with the sixth is of the fourth.

Let A B the first, be the same multiple of C the second, that DE the third, is of F the fourth; and BG the fifth, the same multiple of C the second, that EH the sixth, is of F the fourth. A G, the first together with the fifth, is the same multiple of C the second, that D H, the third together with the sixth, is of F the fourth.

Because A B is the same "ulti

ple of C that DE is of F (Hr), A B contains as many magnitude each equal to C, as DE contains each equal to F. For the same

reason, BG contains as many each equal to C, as EH contains each equal to F. Therefore the whole A G contains as many each equal to C, as the whole DH contains each equal to F. Therefore A G is the same multiple of C that DH is of F; that is, A G, the first and fifth together, Is the same multiple of the second C, that DH, the third and sixth together, is of the fourth F. If, therefore, the first be the same multiple, &c. Q. E. D.

COROLLARY.-From this it is plain, that if any number of magnitudes AB, BG, G H be multiples of another C; and as many DE, EK, KL be the same multiples of F, each of each; the whole of the first,-viz., A H, is the same multiple of C, that the whole of the last,-viz., DL, is of F.

If the first be the same multiple of the second, which the third is of the fourth, and if of the first and third there be taken equimultiples; these are equimultiples, the one of the second, and the other of the fourth.

Let A the first, be the same multiple of B the second, that C the