A Long Way from Euclid by Constance Reid

By Constance Reid

This vigorous advisor via a admired historian makes a speciality of the position of Euclid's Elements in mathematical advancements of the final 2,000 years. No mathematical history past trouble-free algebra and airplane geometry is critical to understand the clear and straightforward factors, that are augmented through greater than eighty drawings. 1963 variation.

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Let be the middle point of CO the shortest distance draw between the two straight lines CD, Through OA, OB parallel to CD, C D and let OX, OY bisect the angle A OB. Take OX, OY,OG for axes of co-ordinates x tana then, if AOB be 2a, the equations of OA, OB are y 2 = 0, and y = x tan a, z 0. , CD . t ; y Hence the equations of the parallel x tan a, z= c; and y = x tan a, s = When it rectangular, tions of CD, W CD Also . lines G are not of importance that the axes should be for axes: the equa jy will then be y = 0, ^ = c and # = 0, z = c.

Cosines are therefore | \/2, |\/2, ^ Ex. 2. Find in a The 3. Am. %(x- Find the direction-cosines of the . Ex. (2, 3, 4) 4. Hence the symmetrical form the equation of the line 3x+y~7z=Q. Ex. direction- actual values of the direction- line 5) x-2y- 5, =y=z- If. whose equations are Ans. -^. Write down the equation of the straight line through the point whicli is equally inclined to the axes. Ans. x-2 = y-3 z-4. THE STRAIGHT 16 LINE. To find the equations of a straight line through two 23. given points.

Acute angle or in the obtuse ? x- x-2y-z= cut ? Is the Is the point (l,-3, 1) in the 1 Ans. cos~ |-v/2, acute, obtuse. Ex. 3. Find the equation of the plane through (1, 4, 3) perpendicular to the line of intersection of the planes 3x + 4y + 7z + 4 0, and + 2z+3 = also of the plane through (3, 1, - 1) perpendicular to the line of intersection of the planes 3x + y-z = 0, 5x-3y + 2z=Q. = Ans. 15x + y-7z + 2=Q. Ex. 4. Shew that the line +iiz+p = Q if IX ^= \ y - -- fj. + m/jL + nv =0, v is parallel to x-y Ans.

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