By Lewis Parker Siceloff, George Wentworth and David Eugene Smith

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**Thirteen Books of Euclid's Elements. Books X-XIII**

After learning either classics and arithmetic on the collage of Cambridge, Sir Thomas Little Heath (1861-1940) used his time clear of his activity as a civil servant to submit many works almost about historical arithmetic, either well known and educational. First released in 1926 because the moment version of a 1908 unique, this publication comprises the 3rd and ultimate quantity of his three-volume English translation of the 13 books of Euclid's components, protecting Books Ten to 13.

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Xy-2y=x2 -;-16. 7. 3 x 2 - xy- 4 x + y = 7. 11. f -x 4 8. -4-= . 12. y 2 (5 10. + x)=- x 8• 13. y (3+x)=x (3-x). 2 2 14. f(x-4)-x2 (x-8)=0. 15. x 2(y+8)+y3 =0. 16. f. 17. 2 :J =9(x2-2x-8). ~-6x+5 18. y = ') x-4 2 """'x- 7 x- 4 19. A and Bare two centers of magnetic attraction 10 units apart, and P is any point of the line AB. P is attracted by the center A with a force P1 equal to 12/A P 2, 10 and by ~e center B with a force F 2 equal ~to 18jBP2• Letting x=AP, express in terms of x the sum s of the two forces, and draw a graph showing the variation of s for all v::Llues of ;::;, 52 LOCI AND THEil~ EQUATIONS 54.

If M divides AB in the ratio 3: 1, and P is a point on AC such that the area of the triangle APM is half the area of ACB, in what ratio does P divide AC? CHAPTER III LOCI AND THEIR EQUATIONS 32. Locus and its Equation. In Chapter I we represented certain geometric loci algebraically by means o£ equations. For example, we saw that the coordinates of the point P (x, y), as this point moves on a circle of radius r and with center at the origin, satisfies the equation x 2 + y 2 = r 2• This is, therefore, the equation of such a circle.

26 GEOMETRIC MAGNITUDES 30. Area of a Triangle. Given the coordinates of the vertices of a triangle, the area of the triangle may be found by a method similar to the one used in surveying. In this figure draw the line RQ through If. parallel to OX, and draw perpendiculars from Oi X I3 and 1j; to R Q. QPz,. Pa = ~(X1Yz- XzY1 + XzYa- XaYz + XaY1- X1Ya)• The student who is familiar with determinants will see that this equation may be written in a form much more easily remembered, as here shown. The area of a polygon may be found by adding the areas of the triangles into which it can be divided.