420 On Fluxions. [Dxc. 
ArticLte VII. 
Further Observations on Fluxions. By Alex. Christison, Esq. 
Professor of Humanity, Edinburgh. 
(To Dr. Thomson.) 
MY DEAR SIR, Edinburgh, Oct.13, 1815, 
To some remarks on Euclid’s definition of proportion, ia his 
fifth book, to be inserted, if you choose, in your Aymals.of Philo- 
sophy, E subjoin the deduction of fiuxions from the definition which 
you published in May. 
if the equality of the products of the means and of the extremes 
be assumed as the criterion of proportionality, it is evident that the 
Ist x A 3d x A 
formula, = ae demonstrates Euclid’s property, in his 
fifth definition, book fifth, with regard both to commensurables and 
to incemmensurables; and that the formula is applicable even to 
abstract numerical and algebraical quantities: but it will be found 
that the criterion above-mentioned is not so convenient as Euclid’s 
for demonstration. We cannot, however, apply Euclid’s criterion 
or property so universally as the other; for we cannot, in demon- 
stration, apply it to alséract numerical and. algebraical quantities : 
we can, apply it to those. quantities only in which, as in geometrical 
magnitudes, the slowest learner sees that the first and the third 
have a necessary dependance on each other, as,also the second and: 
the fourth. A learner understands immediately Euclid’s, definition 
if he be directed to prop. 33, book 6 ; for supposing the angle at 
the centre, the first term is an arch of the one circle, and the third 
term is.the corresponding angle of that arch. Nov it is impossible 
for the slowest learner to, conceive that he can, double, &c. the arch 
or first term. without doubling, &c. the corresponding angle or third 
term. The same may be said with regard to the second and the 
fourth terms, which belong to the other circle. If the one circle 
be laid on the other, and if the multiple of the one arch be equal 
to the multiple of the other, the multiple of the one angle must 
also be evidently equal to the multiple of the other ; and if greater, 
greater ; and if less, less: -consequently the quantities are, by the 
definition, proportional. The equimultiples of the first and third, 
and of the second and fourth, can be exhibited to the learner with- 
out taking any particular numbers as multipliers: but it is impos- 
sible, | think, to do so with regard to any abstract numerical and 
algebraical quantities which are to be proved proportional; and if 
we take parts, we abandon Euclid’s definition, Huclid’s definition, 
then, is not applicable to all proportional, quantities ;, but it is per- 
fect if it be limited, by its proper range: it admits, but it does not 
need, demonstration ; it includes incommensurables; and it de- 
