(3.) INDUCTIVE REASONING BASED ON KNOWN FACTS.-Having already carefully stated the argument* which we wish to present under this head we will here repeat that previous statement: "(a.) The radius of the octant is a natural unit or complete numerical quantity and is necessarily divisible into unital increments, each of them numerically complete in itself, and in quantitative composition similar to the primal unit of which it is an unital increment; therefore the radius of the octant is naturally divisible into ten equal parts, each of which is naturally divisible into ten equal parts; and so on: that is to say, the radius of the octant is decimally divisible, and can be only decimally divided, into unital increments, for the primary unital increment is the one-tenth part, the secondary unital increment is the one-hundreth part, the tertiary unital increment the one-thousandth part, and so on. (b.) Because the radius of the octant is naturally divisible into ten equal parts, the arc and the sine of the octant are each of them also naturally divisible into ten equal parts. (c.) Because the sine and the diff. together compound the arc-length of the octant and because each unital increment of arc contains a primary unital increment of sine together with a primary unital increment of diff., which primary unital of diff. is compounded of a secondary unital increment of sine together with a secondary unital increment of diff., which secondary unital increment of diff. is compounded of a tertiary unital increment of sine, together with a tertiary unital increment of diff., and so on,—therefore when the arc of the octant is evenly divided into ten parts, the magnitudinal difference between the arc and the sine is equal to one of those ten equal parts. Wherefore :-The ratio of the octantal arc to its sine is the ratio of ten to nine." SCHOLIUM. It has been hitherto, we believe, generally, if not universally, assumed that the denary Scale of Number, commonly called the Arabic system of Notation, is a highly ingenious artificial arrangement devised by the human mind for the purposes of computation. We feel sure, however, that attentive consideration of the foregoing demonstration and exposition will result in such assumption being relinquished as no longer tenable. Because it becomes evident, in the first place, that magnitude is fundamentally identical in plan with quantity or number; that is to say, both are grounded on the same unital basis and denary scale; for the "number" or "quantity" ten is a unital increment complete in itself and compounded of ten lesser unitals each of which possesses a like completeness. For example, in the application of quantity to comparative measures of length, the foot and the yard are artificial units: * "Demonstration of the Circle's Ratio," published in 1879. To my droda éremference to ite d'ameter, as letermined by the fireQUEZ GENOMSAubom, may be expressed as // = 3142..., when the diameter 19 9 10 The direct, demonstration, however, is that the octantal are = X 9 when radius equals 1; and, since the octant's sine then = √3, it follows that the perimeter of the inscribed aquare is to the circle's circumference as nine to ten. QUADRATURE This ratio being mathematically established the actual geometrical quadrature becomes a very simple matter. PROP. III. PROBLEM. To describe a Square lineally equal to a given Circle (or vice versa). Let the square e f g h be inscribed in the circle A B C D. Divide one of the sides (as e) into nine equal parts. Bisect one of the nine equal parts and increase the length of each of the four sides of the square one-ninth of itself by addition of half the one-ninth part to each extremity (of each of the sides). The perimeter of the greater square a bed, so constructed, equals, by the foregoing geometrical demonstration, the circumference ABCD of Bbf Л и Tel hd D the circle for the lineal circle and the greater square both have the same ratio to the perimeter of the inscribed square, namely, that of ten to nine, and therefore they are equal. RECTIFICATION.-It has been shown and is mathematically well established that the area of the circle is equal to the circle's circumference multiplied by half the radius, or, what comes to the same thing, to half the circumference multiplied by the radius. We are not aware that it has hitherto been shown, but, if not, it may readily be shown, that the side of a square areally equal to a circle must be a mean proportional between a straight line which equals a fourth part of that circle's circumference and the side of the circumscribed square. For, let radius equal one, then the side of the circumscribed square (or the diameter) equals two, and the fourth part of the circumference multiplied by two equals half the circumference multiplied by one which equals the circle's area. Put a for circle's area, c for circumference. Then obviously, : √a :: √a: 2; because × 1 = a. For the quadrature 4 above, we have shown how to describe a square of which each side equals a fourth part of the circumference of a given circle, and it is now made apparent that, to describe a square areally equal to a given circle, it is requisite to find a mean proportional between a side of the lineal square, which equals the circumference of the circle, and a side of the circumscribed square. The mean proportional will be the side of a square areally equal to the given circle. с с 2 fig. 5. Corollary—'»ree themquare KM NO Fig. 3 le avvally equal to the circle A B C D, Je towa tena mwen of the four angar figure eut off from the square, at the emmers, by the dide's Cremference, areally equaleach of the four segments cut off from the circle by the sides of the square; and, by the same demonstration, the compounded figuresFigs. 4 and 5-are each areally equal to a complete circle, of the circumference of which the curvilineal part of the boundary in the figure is a part, and also each equal to a square of which the base line in Fig. 5 is one of the sides. * Obtained by increasing each side of the inscribed square by one-ninth of itself, as shown for the Quadrature, |