TALBOT8912

Llandaff.
Wednesday evening Dec^r 7^th.

My dear Father.

I have started a plan of making mathematical notes on fools-cap paper under different heads such as cycloid, radical axis, &c, which I think will prove useful. It is the same plan which I adopt for architectural notes.

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I send you a problem.

Required to divide CB. the arc of a circle into any no of p equal parts. Draw ADE t a tangent to the ⊙.

Through C. describe a cycloid whose generating circle = CBD. and whose base is in AD. and let it meet AE in A.

Divide AB into the required no of equal parts & through these pts of division describe other cycloids equal to the 1^st to cut the given arc. The given arc is divided into as required. for it is manifest that the circle may be considered as the generating circle of each cycloid from which it is manifest that arc CB = AB KB = FB. LB = HB.

Q. E D F

Also it is manifest that if a cycloid were cut out in card and pushed with its base always on the line AE. The end of the base might be made to coincide with a point of section of AB. and the cycloid struck to find the pt of section of the arc .CB which would be easy.

We may then inscribe d a figu a rgu rgu any regular polygon in a ⊙. A cycloid may be constructed practically as follows –

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Let .CHD. be the generating ⊙ Take a tangent AB. where .AD = DB and AB is approximately equal to 2πr r being the radius.

divide .AD. into any no of equal parts in K. L. &c.

and the ½ ⊙ .CD. into the same no of equal parts in E. H.

Draw ⊙s = to the original ⊙ to touch .HD. in its pts of secⁿ and draw lines through E H & C || to AD.

Now M where the 1^st ⊙ from A meets the 1^st || from D is a pt on the cycloid

Similarly O where the 2^nd ⊙ meets the 2^nd ||.

∴ by taking a large no of ⊙s & a large no of ||s. and joining the points – AM A. M. O. C. &c we get a practical cycloid & when the no of ⊙s is infinite a true cycloid.

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Now .ABC being a cycloid described by B on the ⊙ce of a rolling ⊙ of what class are the curves. DEF traced by a point .E more remote from the centre than B. and .HRL traced by a point .K nearer the centre, of which class of curves the straight line traced by the centre – and also the cycloid are particular cases?

The straight line through the centre is one limit. One other limit is two straight lines vertically ⊥ to AC through the extremities of the base .AC.

Your affect son
Charles.

P.S. Since when have you taken to writing centre, center? ¹ after the manner of heretics.

P.S.
I have got a method of dividing a circular arc into any number of equal parts, by describing equal cycloids. as the cycloids are equal they may be cut out in cardboard and so struck. Also I have a way of co finding points on a cycloid sufficiently correctly for constructing the curve for practical purposes.

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H Fox Talbot Esq
Athenaeum Club
London