Definition of the third species.

Four numbers are proportional, when the first is a part of the second, and the third is a like part of the fourth.

Definition of the fourth species.

Four numbers are proportional, when the first is a multiple of a part of the second, and the third is a like multiple of a like part of the fourth.

N. B. The above definitions are confined to numbers merely for the sake of brevity; but it is readily perceived, that all the species, except the fourth will admit the unit.

Now, since the aggregate of the definitions of all the species will never define the genus, we are still unacquainted with the generical nature of four proportional numbers. We know, however, that proportionality must here consist of something common to, and necessarily existing in, each of the four spe cies; but whatever this may be, it cannot depend on the divisibility of a unit for this inquiry relates solely to abstract numbers, whose units are void of extension, and, consequently, indivisible. Again, since proportionality here consists of something common to, and necessarily existing in, each of its spe cies, it must, therefore, be something common to, and necessarily existing in, each of the four following

Examples of proportional numbers.

1st species, 7: 7: 13:13.

Numbers are composed of units, but a unit is not a number; if a book be said to consist of leaves, it is plain that a leaf is not a book.

Further, since each of these species is defined by means of a particular relation subsisting in common between the first and second, and between the third and fourth terms, we infer that the genus must be defined by means of some general relation subsisting in common between the first and second, and between the third and fourth, of all proportional numbers. But from inspecting the above examples, it appears that this general relation cannot be expressed by like parts of the first and third, or of the second and fourth terms: for, in the example of the first species, the first term 7 and the third 13, being prime numbers, have no like parts; besides, all the species, except the fourth, admit the unit as a proportional term; and we cannot take a part of a number, and a like part of an impartible unit. We therefore see that a true definition of proportional numbers cannot be formed from the like parts of the first and third, or of the second and fourth terms. Let us next inquire whether the general relation, constituting the proportionality of numbers, can be expressed by means of like multiples of the first and third, or of the second and fourth terms. Here we know that in any four numbers, like multiples of the first and third, or of the second and fourth, may always be taken. Keeping this in view, let us carefully examine the above examples of the four species of proportional numbers. In the first species, the second term 7 is a part of (21) a multiple of the first, and the fourth term 13 is a like part of (39) a like multiple of the third; in the second species, the second term 4 is a part of (24) a multiple of the first, and the fourth term 5 is a like part of (30) a like multiple of the third; in the third species, the second term 18 is a part of (36) a multiple of the first, and the third term 12 is a like part of (24) a like multiple of the third; and in the fourth species, the

second term 12 is a part of (48) a multiple of the first, and the fourth term 6 is a like part of (24) a like multiple of the third. From the general relation here discovered in the four particular examples of the species, we now presume that the following is a true definition of proportional numbers: viz. Four numbers are proportional, when the second is a part of a multiple of the first, and the fourth is a like part of a like multiple of the third. Again, we inquire whether this definition, in its application to the four species of proportional numbers, is universally true. This will appear from the demonstrations of the four following propositions.

PROP. I.

If the first of four numbers be equal to the second, and the third be equal to the fourth, these numbers are proportional.

For, let AB, CD, EF, and GH, represent any four numbers, in which AB-CD, and EF GH; also let ABIK be any multiple of AB, and EFLM a like multiple of EF; then, by inspection, it is evident that CD is a part of (ABIK) a multiple of AB, and that GH is a like part of (EFLM) a like multiple of EF: therefore, by our definition, AB : CD :: EF: GH, as was to be shown..

Corollary. In four proportional numbers, if the first be equal to the second, the third is equal to the fourth.

The following definition has here been rejected as incompatible with the true idea of ratio; viz. Four numbers are proportional when the first is a part of a multiple of the second, and the third is a like part of a like multiple of the fourth.

PROP. II.

If the first of four numbers be a multiple of the second, and the third be a like multiple of the fourth, these numbers are proportional.

For let ABCD, EF, GHIK, and LM, represent any four numbers, in which ABCD is a multiple of EF, and GHIK a like multiple of LM; also let ANOD be any multiple of ABCD, and GPQK a like multiple of GHIK; then it is evident, by inspection, that EF is a part of (ANOD) a multiple of ABCD, and that LM is a like part of (GPQK) a like multiple of GHIK: hence, by our definition, ABCD : EF :: GHIK : LM, which was to be shown.

Corollary. In four proportional numbers, if the first be a multiple of the second, the third is a like multiple of the fourth. PROP. IIL

If the first of four numbers be a part of the second, and the third be a like part of the fourth, these numbers are proportional.

For, let AB, CDEF, GH, and IKLM, represent any

four numbers, in which AB is a part of CDEF, and GH a like part of IKLM; let ANOP be a multiple of AB, and GQRS a like multiple of GH;. then, by inspection, CDEF is a part of (ANOP) a multiple of AB, and IKLM is a like part of (GQRS) a like multiple of GH: hence, by our definition, AB : CDEF :: GH: IKLM, as was to be shown.

Corollary. In four proportional numbers, if the first be a part of the second, the third is a like part of the fourth.

PROP. IV.

If the first of four numbers be a multiple of a part of the second, and the third be a like multiple of a like part of the fourth, these numbers are proportional.

For, let AB, CDEF, GH, and IKLM, represent any four numbers, in which AB is a multiple of a part of CDEF, and GH a like multiple of a like part of IKLM; let ANOP be a multiple of AB, and GQRS a like multiple of GH; then, from inspecting the figure, it is plain that CDEF is a part of (AÑOP) a multiple of AB, and that IKLM is a like part of (GQRS) a like multiple of GH: therefore, by our definition, AB : CDEF :: GH: IKLM, which was to be shown.

Corollary. In four proportional numbers, if the first be a multiple of some part of the second, the third is a like multiple of a like part of the fourth

Since, therefore, we have found our definition to be universally correct, this inquiry is finished.