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

Definition XII.

IF the right line BA (num. 1. 109.) one end at B remain∣ing fixed, be moved round at the other end with an equal motion from A thro' C, D, E to A back again, and in the mean while, there be conceived another moveable point in it to move with an equal motion along the line BA from B to A, so that in the same moment wherein the moveable point A absolves one revolution, the other moveable point shall also have passed thro' its right lined way, coinciding with the point A retur∣ned to its first situation; this extremity A by its revolution will describe the circle ACDEA, and that other moveable point another curve B, 1, 2, 3, 4, &c. which with Archi∣medes we will call a Helix or spiral Line, and the plane space comprehended under this spiral line and the right line BA in the first station is called a spiral space. Now if we suppose, e. g. the right lined motion of the point moving along BA to be twice slower than in the former case, so that (see num. 2.) in the same time that the point A makes one whole Re∣volution, the other moveable point shall come to F, making half the way BA, and then at length shall concur or meet with the extremity, when that shall have performed the other revolution; and so there will be described a double spiral line, whose parts with Archimedes we will so distinguish, that as he calls the part of the right line BF, passed over in the first revolution, simply the first line, and the circle made by the right line BF the first Circle; so we will call that part of the curve which is described in that time or revolution B 136912 the first Helix or the first Spiral, and the area comprehended un∣der it the first spiral space: And, as the other part of the right line FA passed over in the other revolution is called the second line, and the circle marked out by the whole line BA the se∣cond Circle; so the curve described in the mean while 12, 15, 18, 24, may be called the second spiral line, and the space

comprehended under it the second spiral space, and so onwards From these Definitions there flow the following

CONSECTARYS.

I. THE lines B 12, B 11, B 10, &c. drawn out, and making equal angles to the first or second spiral (and after the same manner(α) 1.1 B 12, B 10, B 8, &c. or B 12, B 9, B 6, &c.) are arithmeti∣cally proportional, as is evident.

II. The lines drawn out to the first spiral as B 7, B 10, &c. are one among another as the arches of the circles inter∣cepted between BA and the said lines(β) 1.2 B 7, B 10, &c. which is also evident to any one who considers what we did suppose; for in the same time as the end A passes over seven parts of the circle, the other moveable point will also run over seven parts of the right line BA, &c.

III. Lastly, The right lines drawn from the initial point(γ) 1.3 B to the second spiral e. g. B 19 and B 22 (num. 2.) will be one to ano∣ther as the aforesaid arches together with the whole periphery added to both sides: for at the same time the extremity A runs thro' the whole circle or 12 parts and moreover 7 parts (i. e. in all 19 parts) in the same time the other moveable point passes through 12 parts of the right line BA (in this case di∣vided into 24 parts) and moreover 7 parts, that is, in all 19; and so in the others.

Notes

• (α) 1.1

  Archim. Prop. 2. of Spirals.
• (β) 1.2

  Archim. Prop. 4.
• (γ) 1.3

  Archim. Prop. 15.