Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.

CONSECTARY IX.

AS Consect. 8. supplies us with a rule to determine the pro∣portion of every spiral of every order to the periphery of the correspondent circle, viz. if the number of the order be doubled for the periphery of the circle, and the next antece∣dent odd number be taken for the spiral line; so what we have hitherto demonstrated supplys also another rule, to define the proportion of the spiral space in any order to its circle. For since the circles are in a progression of Squares 1, 4, 9, 16, &c. but the first circle is to the first space as 3 to 1 (i. e. a, 1 to ⅓) by Prop. 17. and the second to the second as 12 to 7 (i. e. as 4 to 2 ⅓) by Prop. 18. the third to the third as 27 to 19 (i. e. as 9 to 6 ⅓) by Schol. 1. of this. And contem∣plating both these series one by another,

• Of the circles, 1, 4, 9.
• Of the spaces, ⅓, 2 ⅓, 6 ⅓.

We see the numbers of the spaces are produced, if from the spuare numbers of the circles you substract their roots, and add to the remainder ⅓. Wherefore, if, e. g. we were to de∣termine the proportion of the fourth circle to the fourth spiral space; the square of 4 viz. 16 would give the circle; hence substracting the root 4, there wiill remain 12, and adding ⅓ you would have the fourth spiral space 12 ⅓; and in like man∣ner the spiral space 20 ⅓ would answer to the circle 25, &c. And that this is certain is hence evident, that if we mul∣tiply these numbers 16 and 12 ⅓, also 25 and 20 ⅓ by 3, that we may have those proportions in whole numbers, 48 and 37, 75 and 61, these are those very numbers Archimedes had hinted at in the Coroll. of Prop. 25.

[illustration]

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