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,

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.

S

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

[r, 8] [nr, ns].

Hence the relative multiple scale of the two numbers r, s determines the rational fraction which is denoted by the symbol.

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.

r
S

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

another,

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′].

Art. 46. Def. 6. MEASURE.

If a magnitude A contains another magnitude B an exact number of times, B is said to be a measure of A.

Art. 47. Def. 7. COMMON MEASURE.

If the magnitudes A and B each contain another magnitude G an exact number of times, then G is said to be a common measure of A and B.

Art. 48. Def. 8. COMMENSURABLE MAGNITUDES.

If two magnitudes have a common measure they are said to be commensurable.

ENUNCIATION 1. The scale of two commensurable magnitudes is the same as that of two whole numbers.

ENUNCIATION 2. Commensurable magnitudes are to one another in the ratio of two whole numbers.

Let A and B be two commensurable magnitudes.

Let N be their common measure.

where a, b are some two whole numbers.

It will now be proved that [A, B] ≈ [a, b].

Take any integers r in the first column, s in the second column of the scale of A, B.

Then there are three alternatives represented by the figures

These are represented in the scale of a, b by the figures

Art. 50. The above proposition expresses the fact that

it is seen that magnitudes in either are replaced by whole numbers in the other.

Art. 52. EXAMPLE 18.

Prove the converse of Prop. 11, viz.:

If the scale of two magnitudes is the same as that of two whole numbers, then the magnitudes are commensurable.

H. E.

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