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

SCHOLIUM V.

THis Theorem of Pythagoras as it furnishes us with Rules of adding Squares into one Sum, or subtracting one Square from another; so likewise it helps us to some Foundations where∣on, among the rest, the structure of the Tables of Sines relies, &c. Whose use we have already partly shewn in Schol. 1 and 4. 1. If several Squares are to be collected into one Sum, having joined the Sides of two of them so as to form a Right Angle, e. g. AB and BC (Fig. 68. N^o 1.) the Hypothenuse AC being drawn, is the Side of a Square equal to them both; and if this Hypothenuse AC be removed from B to D, and the Side of the third Square from B to E, the new Hypothenuse DE will be the Side of a Square equal to the three former taken together. 2. If the Square of the side MN (N^o 2.) is to be subtracted from the Square of the side LM. Having described a Semicircle upon LM, and placed the other MN within that Semicircle, then draw the Line LN and that will be the Side of the remain∣ing Square. 3. Having the Right Sine EG of any Arch ED gi∣ven (but how to find the Primary Sines we will shew in another place), you may obtain the Sine Complement CG or EF, by the preceding

Numb. viz. by subtracting the □ of the given Sine from the □ of the Radius; and moreover the versed Sine GD by subtract∣ing the Sine Complement CG from the Radius CD. 4. The Squares of the versed Sine GD, and of the Right sine EG being added together, give the □ of the Chord ED of the same Arch, (which all are evident from the Pythagorick Theorem) and half of that EH gives the Right Sine of half that Arch. 5. From the Right Sine EG you have the Tangent of that Arch, if you make, as the Sine Complement CG to the Right Sine GE, so the whole Sine CD to the Tangent G I. 6. Lastly, From these Data you may also have the Secants (if required) thus, as the Sine Complement CG to the W. S. CE, so the W. S. CD to the Secant CI; or as the Right Sine EG to the W. S. E.C. so the Tangent ID to the Secant IC; both which are e∣vident by our 34th Proposition.

Consect. 9. If the Quadrant of a Circle (CBEG, Fig. 70.) be inclined to another Quadrant (CADG) and two other Per∣pendicular Quadrants cut both of them, viz. FBAG and FEDG, and the latter do so in the extremities of them both) having let fall Perpendiculars from the common Sections E and B, thro' the Planes of the Perpendicular Quadrants, and the inclined Quadrant, (viz. on the one side EG and BH, as Right Sines of the Segments EC and BC; on the other EI and BK, as Right Sines of the Segments ED and BA) you'l have 2 Tri∣angles EIG and BKH Right Angled at I and K, Equiangular at G and H (by reason of the same inclination of the Plane CBEGC) and consequently similar, by our 34th Proposition; wherefore as the Sine EG to the Sine EI, so the Sine BH to the sine BK, or as EG to BH so EI to BK, and contrariwise.