These numbers are found thus: a′ is the geometric mean between a and A; that is, between .5 and .25, and A′ is the arithmetic mean between a′ and A, or between .3535533 and .25. Again, a′′ is the geometric mean between a′ and A′; and A′′ the arithmetic mean between a′′ and A′. Proceeding in this manner, we find a(13) = .3183099; A(13) = .3183099. Hence the area of a circle radius unity, correct to seven decimal places, is equal to the reciprocal of .3183099; that is, equal to 3.1415926; or the value of π correct to seven places of decimals is 3.1415926. The number π is of such fundamental importance in Geometry, that mathematicians have devoted great attention to its calculation. Mr. Shanks, an English computer, carried the calculation to 707 places of decimals. The following are the first 36 figures of his result:— The result is here carried far beyond all the requirements of Mathematics. Ten decimals are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimals would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful microscope. CONCLUSION. In the foregoing Treatise we have given the Elementary Geometry of the Point, the Line, and the Circle, and figures formed by combinations of these. But it is important to the student to remark, that points and lines, instead of being distinct from, are limiting cases of, circles; and points and planes limiting cases of spheres. Thus, a circle whose radius diminishes to zero becomes a point. If, on the contrary, the circle be continually enlarged, it may have its curvature so much diminished, that any portion of its circumference may be made to differ in as small a degree as we please from a right line, and become one when the radius becomes infinite. This happens when the centre, but not the circumference, goes to infinity. THE END. THIRD EDITION, Revised and Enlarged—3/6, cloth. _____ A SEQUEL TO THE FIRST SIX BOOKS OF THE ELEMENTS OF EUCLID. BY JOHN CASEY, LL.D., F.R.S., Fellow of the Royal University of Ireland; Vice-President, Royal Irish Academy; &c. &c. ________________ Dublin: Hodges, Figgis, & Co. London: Longmans, Green, & Co. ________________ EXTRACTS FROM CRITICAL NOTICES. “Nature,” April 17, 1884.