THE(α) 1.1 first spiral space is subtriple of the first circle, i. e. as 1 to 3.
Having divided the circumference of the circle into (Fig. 141. n. 1.) three equal parts by lines drawn from the initial point, beginning from the first line BA, the line BC will be as 1, BD as 2, BA as 3, by Consect. 1. Def. 12. of this book, and consequently the sectors circumscribed about the spiral will be CBc as 1, DBd as 4, ABa as 9, by Prop. 32. lib. 1. and in like manner, if you make new bisections, the lines drawn from the point B to the spiral, will be 1, 2, 3, 4, 5, 6; but the circumscrib'd sectors, 1, 4, 9, 16, 25, 36; and so the circumscrib'd partial sectors ad infinitum will proceed in an order of squares, there being always as many sectors in the circle equal to the greatest of them. Therefore all the sectors that can be circumscrib'd ad infinitum about the spiral space, i. e. the spiral space it self (in which at last they end) to so many equal to the greatest, i. e. to the circle, is as 1 to 3, by Consect. 10. Prop. 21. lib. 1. Q. E. D.
SInce the first circle is to the second as 1 to 4 (i. e. as 3 to 12) by Def. 12. of this, and Prop. 31. lib. 1. and the first spiral space to the first circle as 1 to 3 by the present Prop. the same spiral space will be to the second circle as 1 to 12; and to the third by a like inference as 1 to 27, to the fourth as 1 to 48, &c.