# Question from Whiteside V. 1 - what did Newton mean by a:b :: c:d notation?

I'm reading volume 1 of Whiteside's 'Mathematical Papers of Isaac Newton,' and on pp. 383-384, Newton reaches a conclusion on his "Example 1st" in the statement "55:-54 :: p:q..." - which seems to be backwards if we are meant to read "::" as "as". But it's also possible that Newton intends "::" to read "is reciprocal to" - which would make his conclusion correct.

This is not a life-and-death matter, just putting this out there if someone has background or interest in Newton-notation.

• Mar 16, 2022 at 7:58

Often the uses of the double colon sign indicate an 'analogy', much like a predecessor of an equation. I haven't looked hard at your example, but unless there's something unusual about it, the use of two of them could be understood as indicating three equal ratios, thus

55 : -54  ::  p : q ::  vel_A : vel_B

55 / -54   =  p / q  =  vel_A / vel_B .


One point of difference between the old and new usages is that traditionally proportions were only regarded as legitimately formed if their members were of like kind, we might say similarly dimensioned.

The old tradition and theory of proportion dating back to Euclid is explained for example by N Guicciardini in "Mathematics and the New Sciences", chapter 8, (esp. sec. 8.3.2, p.232 et seq.) in 'Oxford Handbook of the History of Physics' ed J Buchwald et al., 2013, see also here.

• Also, the older approach, namely, proportions versus fractions, did not blow up when "denominators" were $0$. Mar 16, 2022 at 2:57
• Thank for the comments. Yes, this is a three ratio expression. And, to your point, all of this came up as I was writing a paper discussing how Galileo and Newton observed - and didn't observe - Euclidian restrictions on ratios. My conclusion is that Newton just got p and q reversed in this draft and that Whiteside didn't happen to catch this one. (These are absolutely wonderful books, btw.) thankas again. TB Mar 19, 2022 at 19:18