therefore, very evident that, should we measure the extent of No. x 13 as a boundary, it would thus be treated as a larger circular boundary than No. x 14 to the extent of No. x 15, and would thus give an area of more than twice the extent of cal. No. × 15, multiplied by one, in excess of what its angular boundary actually contains. For if half the boundary multiplied by half the diameter gives the true area of a circle, consequently the whole boundary multiplied by one-fourth the diameter will give the same amount. But in this case the diameter would also be enlarged in proportion to the greater extent of the boundary; to find the value of this increase as a proportional, say as the true boundary, per cal. No. x 14, is to the diameter of 8 inches, so is the boundary of cal. No. x 13 to the diameter required for it; and one-fourth the extent of the diameter thus found, when multiplied on cal. No. ×13, will give the amount of its supposed circular area; and this being so greatly in excess of the area which it actually encloses, clearly proves the utter absurdity of measuring it as a boundary to find its true area. Having premised these facts, I will now proceed with my investigation as to the merits of the afore-mentioned polygon boundary, supposed to have been obtained from the linear measure of the angular surfaces of a numerously-sided polygon, assumed to be constructed within the area of the true circular boundary, so that each minute side of said polygon is thus regarded as forming a separate and distinct chord of a corresponding arc or segment of their enclosing boundary. But the area of those segments, and the extra boundary measure of their curvature, is regarded as being cut off, or at least not included in either the boundary or area of the polygon measurement. Yet it must be admitted as an axiom, that both the area and boundary of the enclosed polygon must necessarily be less than that of its surrounding circle, which includes those segments. Now, the first thing to ascertain is the simple fact—has the polygon, as stated, been actually formed within the area of the true circular boundary? This is very easily found by taking, say, Van Cullen's boundary of the circle for a diameter of 2 inches, which, on being multiplied by 4, will give the proportional boundary for a diameter of 8 inches; and comparing this boundary with that given as the true boundary for 8 inches diameter, per cal. No. × 14, I have chosen to compare this with the Van Cullen boundary, although I am aware that the ten-decimal product of the inscribed and circumscribed polygons is that which is generally adopted in works on circular and spherical geometry. But as the Van Cullen boundary of 35 decimals agrees with the former to the extent of 9 of the said 10 decimals, it will thus attest the correctness of both to that extent, which also represents the greater part of all the difference. CIRCULAR BOUNDARIES. Van Cullen's polygon boundary...=25 132741228 71834590770114706623602304 for 8 in. dia. ... It is therefore manifest that the Van Cullen polygon was not constructed within the true circular boundary; as, in such case, the chords formed by its sides could not possibly have extended beyond their circular enclosure, nor be even equal to it in their linear extent. But, on the contrary, we find them extending considerably beyond it, and even including a much larger area than the circle itself, as appears by the above figures, that is, if they be regarded or measured for area as a circular boundary. For, as the true boundary for 8 inches diameter when multiplied by 2 inches, or one-fourth the diameter, will give the true area, consequently the above boundary, when multiplied by onefourth of a diameter proportional thereto, would give an area considerably greater even than the above figures multiplied by 2 would yield, and the excess would also show more than twice the amount above stated. MEASURE OF THE AREAS OF CIRCLES. V.C.'s supposed area of circle=50 265482457,43669181540229413247204608 for 8 in. diameter. True circular area Difference in excess ... ... =49.883063257 98366605359045463536912416 = 382419199 45302576181183949710292192 I have also examined the area of the inscribed polygon of 10 decimals, as given in "Leslie's Geometry," page 355, for an assumed radius of 1 inch, or half the area of a circle of 2 inches diameter. And having increased this area to a proportional for 8 inches diameter, and taking one-half of the linear extent of the figures representing that amount for the measure of the outer sides of the polygon, I then divided this extent by the given number of sides to obtain the measure of a single side, which, on being multiplied by the height of the triangle underneath valued as 4 inches, and taking half the amount for its area, gave exactly a true proportional of the value given for the whole area of the polygon as stated, which, had it been multiplied by any less amount, it could not have done. It therefore follows that the height given for the common radius of the inscribed and circumscribed circles, which is stated to be considerably less than the height of the centre of a side of the inscribed polygon, cannot be correct as a proportional of the area given for that of its total contents. It thus appears very evident that it would require a radius not less than 4 inches to form an inscribed circle only touching the centres of the respective sides of a circumscribing polygon of the area given, as clearly appears by the figures stated on page 21; and the value of the angles thus cut off the area of the polygon by the inscribed circle is also given in the figures just referred to, as so much in excess of the true circular area. Now, without pursuing this enquiry farther, I think what I have already brought forward fully justifies me in stating that neither the surface measure of the circumscribing polygon nor its area can be regarded as a true proportional of the diameter given, which only applies to a perfect circle, and not to the polygon. In confirmation of these facts, I think that it is only necessary to give "Euclid's" definition of what constitutes a circle and its diameter "A circle is a plane figure bounded by one line, which is called the circumference; and is such, that all right lines drawn from a certain point within the figure to the circumference are equal to one another, and this point is called the centre of the circle. A diameter of a circle is a right line drawn through the centre, and terminated both ways by the circumference."-Defs. 15, 16, and 17, B. 1. Now, a polygon is not bounded by one line, nor are all right lines drawn from its centre to the circumference equal to one another; not being equidistant from the centre they cannot be, and consequently the true circle diameter, as described, cannot be applied to the area of a polygon, nor the boundary of the latter to that of the perfect circle. In conclusion, I would simply note that I consider the similarity in value of the boundaries and areas appearing in both polygons merely indicates the idea that their respective authors were equally deceived by the mode adopted by each to obtain their figures; while entertaining the supposition that they could actually place within the six arcs formed under a circumscribing circle around a hexagon nearly the entire area of six squares, within a space only capable of containing four of like magnitude. This appears in the excess of their boundaries and areas over that of the true circle, which represents nearly one-third of the area of their included squares, evidently so much in excess of what they were intended to occupy. The explanation of the errors stated will appear very plain on reference to page 13, where may be seen my reasoning, and the mode explained by which I was enabled to obtain the true circular boundary. The mode of reckoning by the common principles of arithmetical computation adopted in this little work, has been specially intended to render the enquiry more simple for the perusal and consideration of the large portion of intelligent people to whom this form of investigation will be familiar. And I have carried out the work so as to save trouble to the reader, as far as possible. All the calculations referred to in the book will be found already prepared with the most perfect accuracy; even the squares are given, and their roots extended in the circular ring part to 80 and in the angular part to 82 decimals. I have also given the diagrams or figures at their full size, that they may more perfectly speak to the eye, while the calculations demonstrate the facts to the understanding. In conclusion, I would now, as an Irishman of the Province of Ulster, desire to submit the result of my investigations (as a small contribution to geometrical science) to my country, and the great nation of which I am proud to be a member; and also to scientific societies generally, soliciting their kind and careful examination of the same. Marcus Ward & Co., Limited, Printers, Belfast. |