then if there are any values of the integers r, s which make

(5) rA=sB, they must also make_rC = sD;
(6) rCsD, they must also make rA = sB;

and therefore the scale of A, B is the same as that of C, D.

Suppose if possible that some values of r, s exist which make rA =sB, but rC not equal to sD.

Consequently rC is neither greater nor less than sD.

Consequently the scale of A, B is the same as that of C, D.

Art. 38. EXAMPLES.

12. If for a single value of the integer r, say r1, and a single value of the integer s, say s1, it is true that

and

r1A=s1B
r1C = 81D,

then prove that any values of the integers r, s which make

Hence in the scale of A, B, the number r in the first column is always on the same level as the same number r in the second column. Hence the scale of A, A is

Art. 40. EXAMPLES.

14. Form the relative multiple scale of

(1) lines 7 inches long and 8 inches long respectively,

(2) lines 8 inches long and 9 inches long respectively.

How far are the two scales the same?

15. Let A be the side of a square, and B its diagonal; form the relative multiple. scale of A and B far enough to show that the tenth multiple of B lies between the fourteenth and fifteenth multiples of A.

16. Let A be the side of a square, and B be the hypotenuse of a right-angled triangle, one of whose sides is A and the other is the diagonal of the square. Form the relative multiple scale of A and B far enough to show that the tenth multiple of B lies between the seventeenth and eighteenth multiples of A.

17. Let A be the diagonal of a square, and B be the hypotenuse of a right-angled triangle, one of whose sides is A and the other is a side of the square. Form the relative multiple scale of A and B far enough to show that the tenth multiple of B lies between the twelfth and thirteenth multiples of A.

SECTION II.

THE SIMPLER PROPOSITIONS IN THE THEORY OF RELATIVE MULTIPLE SCALES WITH GEOMETRICAL APPLICATIONS. FIRST SERIES. Nos. 9-16.

Art. 41. PROPOSITION IX.* (Euc. V. 15.)

ENUNCIATION 1. To prove that the scale of A, B is the same as the scale of nA, nB,

ENUNCIATION 2. To prove that two magnitudes have to one another the same ratio as their equimultiples,

Take any integers r, s in the first and second columns respectively of the scale of A, B.

There are three alternatives, represented by the figures

which are represented in the scale of nA, nB by the figures.

Comparing the Figures 22, 23, 24 with the Figures 19, 20, 21 respectively, it follows at once that

the scale of A, B is the same as that of nA, nB*.

Art. 42. The case of the above Proposition in which n = 2 will often be required, i.e.

[A, B] ~ [2A, 2B]:

Art. 43. Since n represents any whole number whatever, it may have an infinite number of values.

Hence nA and nB represent an infinite number of pairs of magnitudes, e.g. 2A and 2B, 3A and 3B,...... such that the scale of any pair is the same as that of A, B.

Hence there are an infinite number of pairs of magnitudes which have the same scale.

Hence if a scale be given, the magnitudes of which it is the scale are not given.

Thus two magnitudes of the same kind determine a definite scale; but if a scale only be given, the magnitudes of which it is the scale are not given.

Art. 44. ARITHMETICAL APPLICATION OF PROPOSITION IX.

Let r and s be two whole numbers.

Let the number r be divided into s equal parts, and let each part be denoted. by the symbol.

* If A and B are numbers, then denoting numbers by small letters it follows that

[r, s] = [nr, ns].

Hence the relative multiple scale of the two numbers r, s determines the

Although the term "ratio" has not yet been defined it may here be stated that the rational fraction is taken to be the measure of the ratio of r to s.

Art. 45. PROPOSITION X. (Euc. V. 11.)

ENUNCIATION 1.

i.e. if and if

then

If the scale of A, B is the same as that of C, D; and if the scale of A, B is the same as that of E, F; then the scale of C, D is the same as that of E, F; [A, B] = [C, D],

ENUNCIATION 2. Ratios which are equal to the same ratio are equal to one

There is a certain scale, viz.:—that of A, B.

The scale of C, D consists of the same arrangement of numbers as that of A, B. So also does the scale of E, F.

Hence the scale of C, D is the same arrangement of numbers as the scale of E, F.

.. [C, D] ~ [E, F′].