(643)
Tables, To 3010299956639811952405804, the Logarithm of 2 in the Tables.
By this means the Area of one Hyperbola being computed, the Area's of all others may be thence argued, as is shewed by Greg. St. Vincent, and Van Schooten in Tractatu de Organica Conicarum sectionum descriptione.
If the Logarithm of 1 be put 0; and of 10, 1,0000000: If between 1 and 10 you conceive 9999999 mean Proportionals interjected; the first is 1,00000023025853.
If the Logarithm of 1 be put 0; and of 10, 100000: If you conceive 99999 mean Proportionals between 1 and 10, the first is 1,0000235853; if an infinite rank of these be continued, there is no number proposed, but will go nigh to be found in this rank, and the number of Terms, by which it is removed from Unit, is the Logarithm of the Number so found. The Ratio of 1, to 1,00000023025853, some call Elementum Logarithmicum. See Cavallieri's Trigonometry.
The Area of an Hyperbola is frequently required in Gauging; as admit it were required to compute the Solidity of the Segment of an upright Cone cut by a Plain, that would cut the produced opposite Cone; in any such Case the Section is an Hyperbola. But we will only take the Instance, when it is parallel to the Axis.
In Figure II.1. Let BVA represent such a Cone, VC its Avis, BSAR the Circle in the Base. And first, suppose this Cone cut by a Plain passing through the Vertex and the Base USRU; then is the whole Cone divided into such proportions as the Area of the Circle in the Base. Whence we discover the use and the want of a good Table of Area’s of Segments; the best of which kind yet extant is in Sibrand Hantz his Century of Geometrical Problems, translated out of Dutch into English by Captain Thomas Rudd, who omitted the said Table; useful likewise for finding the Area of the Segment of an Ellipsis, and the obtaining the quantity of Liquor out of, or left in a Cask part empty.
And we hint, that a Table of Natural Versed Sines is to be found in Maginus, and of Logarithmical ones in Cavallieri's Directorium Universale Uranometricum.
2. The former Plain did cut out a Chord-line in the Base, to wit, SR; through the same imagine another Plain to pass, and to cut the Cone beneath the Vertex, as at O:, then is the Wedge contain'd between both these Plains (to wit, VSROV) equal to 13 of that Cylindrick or Prismatick Figure, whose Altitude is equal to the Perpendicular VP falling from the Vertex of the Cone to the cutting Plain, and whose Base SORTS is the Area of the Figure cut; in this case, an Hyperbola: When the Plain passeth parallel to the side BV, a Parabola; when it will meet with VB produced, a Portion of an Ellipsis. By this means, if a Brewer's Tun (taken to be a Circular Truncus Coni) lean, and be not cover'd over with Liquor in its bottom, it may be computed by subtracting the two known before-men-
