the superincumbent mass of the wall above. The two remaining pillars, derived from the hyperbolic and conchoidal[12] frusta, are nearly identical in form; and of these two curves I preferred the former, which gives the most advantageous arrangement of materials, in regard to stability, of all the four forms. This quality of advantageous proportion exists in these forms, in the ratio of the numbers in the last column of the following table:[13] which shews a slight superiority of the Hyperbolic over any of the other forms. +------------+------+-----------+--------+---------+-----+----------+ | | | | Volume | Distance| | | | |Height| Diameter | of | of | | | | |of the| | | solid |centre of| | Economic | | | Tower| at | at | Tower | Gravity | |Advantage.| | | in | Base| Top |in cubic| from | H | G·M. | |Hypothetical| feet.| in | in | feet. | Base. | - | ------ | | Towers. | (H.)|feet.|feet.| M. | G. | G | G′·M′. | +------------+------+-----+-----+--------+---------+-----+----------+ |Hyperbolic, | 120 | 42 | 16 | 62,915 | 41·227 |2·911| 1·00000 | |Conchoidal, | 120 | 42 | 16 | 62,984 | 41·336 |2·903| 0·99627 | |Parabolic, | 120 | 42 | 16 | 63,605 | 43·400 |2·765| 0·93963 | |Logarithmic,| 120 | 42 | 16 | 74,742 | 42·460 |2·826| 0·81608 | |Conical, | 120 | 42 | 16 | 84,737 | 43·280 |2·773| 0·70725 | +------------+------+-----+-----+--------+---------+-----+----------+ [12] The solid, in this case, would have been formed by the revolution of the interior conchoid of Nicomedes about its directrix; and its co-ordinates were kindly calculated for me by my late revered preceptor, Dr WALLACE, Professor of Mathematics in the University of Edinburgh, who employed so many hours of his latter years in labours of kindness among his friends. This act of the Professor was the result of a conversation I had with him on the subject. Before I received his friendly communication, however, I had resolved to adopt the rectangular hyperbola, whose co-ordinates I had myself determined with this view some time before; and when I found that the conchoid and the hyperbola, traced between the two fixed points by means of the calculated co-ordinates, were so nearly coincident, that it was difficult to prevent their running into each other, even when drawn out on a large scale, I determined to adhere to my original purpose of adopting the latter curve as my guide. [13] The last column of this table is derived as follows:--Assuming that the economic advantage of any proposed tower of given height and diameter at base and top, is _inversely_ as the mass and the height of the centre of gravity above the base, and denoting these quantities by M and G respectively, the fraction 1 --- G·M