We are not sure if Mr. Isaac Newton of Wisbech was a relative of the polymath of Woolsthorpe Manor, or if he is a nom de plume (as a sort of joke). However, his solution to the scale problem, below, is very learned if a little daunting!
“To the Editor of the Mercury.
Wisbech, April, 1824.
Sir,
As scientific subjects sometimes have place in you columns, I choose to send you the following. It is a problem of perpetual use among Land-Surveyors; and as the method which I have here given of performing it, appears to me more simple, convenient, and accurate, than any which I have hitherto met with, I have no doubt that a perusal of it will prove interesting to some of your scientific readers. The problem is-
‘To transfer any map or plan of a given scale, to another scale, which shall be given integral number of times smaller or larger than the given scale.’
Solution.- Let the rectangular parallelogram ADCB represent the given map or plan, and let E, F, G, &c. be the points, places, or objects, depicted thereon. Bisect A B the base of the parallelogram in M, and make the perpendicular M P, below A B, equal to A D or B C, the adjacent side to A B: drawe the right lines P A, P D, P C, P B, P E, P F, P G, &c., in which lines take the points a, b, c, d, e, f, g, &c. respectively; making P A to P a, P D to P d, P C, to P c, P B to P b, P E to P e, P F to P f, P G to P g, &c., as the given scale to the required one; and draw the right lines a d, d c, c b, b a, so shall the figure abcd, containing the points e, f, g, &c. be the required plan; being similar to the given one, and on a scale a given integral number of times smalled than the given scale.
When the required plan is to be on a scale, a given integral number of times smaller than the given scale, it is only necessary to project P A, P D, P C, P B, P E, P F, P G, &c. at the extremities A, D, C, B, E, F, G, &c. and proceed as in the former case. In both cases whatever lines lie between the points, E, F, G, &c. in the given plan, the same or similar lines must also connect the corresponding points in the required plan, which will then be complete. – I am, &c. Isaac Newton.”
The Stamford Mercury, 16th April, 1824.
(We are indebted to John Riley for grappling with this problem and drawing the example above, which better demonstrates what this is all about!)