The mathematician meets with some. thing extraordinary in Sharp's elaborate " Treatise of Geometry Improved," (in 4to. 1717. signed A. S Philomath,) 1st. by a large and accurate table of segments of circles, its construction, and various uses in the solution of several difficult problems, with compendious tables for finding a true proportional part, and their use in these or any other tables exempli fied in making logarithms, or their natu ral numbers to 60 places of figures, there being a table of them for all primes to 1100, true to 61 figures. 2d. His consise " Treatise of Polyedra," or solid bodies of many bases, both the regular ones and other ; to which are added twelve new ones, with various methods of forming them, and their exact dimensions in surds or species, and in numbers; illustrated with a variety of copper-plates, neatly engraved by his own hands. Also, the models of these polyedra he cut out in box-wood, with amazing neatness and accuracy. Indeed few or none of the mathematical instrument-makers could exceed him in exactly graduating or neat ly engraving any mathematical or astro nomical instrument, as may be seen in the equatorial instrument above mention. ed, or in his sextant, quadrants, and di. als of various sorts ; also in a curious ar millary sphere, which, beside the com mon properties, has moveable circles, &c. for exhibiting and resolving all spheri cal triangles; also his double sector, with mare other instruments, all contrived, graduated, and finished by himself. In short, lie possessed at once a remarkably clear head for contriving, and an extra ordinary hand for executing any thing, not only in mechanics, but likewise in draw ing, writing, and making the most exact and beautiful schemes or figures, in all his calculations and geometrical construc tions.
The quadrature of the circle was un dertaken by him for his own private amusement in the year 1699, deduced from two different series, by which the truth of it was proved to 72 places of fi gures; as may be seen in the introduction to Sherwin's table of logarithms ; that is, if the diameter of the circle be I, the cir cumference will be found equal to 3 14 1592653589793238462643383.795u28841 971693993751U5820 749445923078164 •5, &c. In the same book of Sherwin's may also be seen his ingenious improvements on the making of logarithms, and the con structing of the natural sines, tangents, and secants.
He also calculated the natural and loga rithmic sines, tangents, and secants, to every second in the first minute of the quadrant ; the laborious investigation of which may probably be seen in the ar chives of the Royal Society, as they were presented to Mr Patrick Murdock for that purpose ; exhibiting his 'very neat and accurate manner of writing and ar ranging his figures, not to be equalled perhaps by the best p. nman now living.
The late ingenious Mr. Ssneaton says, (Philosophical Transactions, anno 1786, p. 5, kc.) : " In the year 1689, Mr. Flam steed completed his mural arc at Green wich; and, in the Prolegomena to his Historia Ccelestis, he makes an ample ac knowledgment of the particular assist ance, care, and industry of Mr. Abraham Sharp ; whom, in the month of August 1688, he brought into the observatory as his amanuensis ; and being, as Mr. Flam steed tells us, not only a very skilful ma thematician, but exceedingly expert in mechanical operations, he was principally employ ed in the construction of the mural arc ; which in the compass of fourteen months he fimshed, so greatly to the sa tisfaction of Mr. Flamsteed, that lie speaks of him in the highest terms of praise.
"This celebrated instrument, of which he also gives the figure at the end of the Prolegomena, was of the radius of 6 feet, 74 incites; and, in like manlier as the sextant, was furnished both with screws and diagonal divisions, all which wire made by the accurate hand of Mr Sharp. But yet, whoever compares the different parts of the table for conversion of i he re volutions, and parts of the screw belong ing to the mural arc, into degrees, mi nutes, and seconds, with each other, at the same distance from the zenith on different sides, and with their halves, quarters, &c.
will find as notable a disagreement of the screw-work from the band divisions, as had appeared before in the work of Mr. Tompion ; and hence we may conclude, that the method of Dr. Hook, being ex ecuted by two such masterly hand, as Tompion and Sharp, and found defective, is in reality not to be depended upon in nice matters.
From the account of Mr. Flamsteed it appears also, Mr. Sharp obtained the zenith point of the instrument, or line of collimation, by observation of the ze nith stars, with the face of the instrument on tie east and on the west side of the wall; and that having made the index stronger (to prevent flexure) than that of the sextant, and thereby heavier, he contrived, by means of pulleys and ba lancing weights, to relieve the hand that was to move it from a great part of its gravity. Mr. Sharp continued in strict correspondence with Mr. Flamsteed as long as he lived, as appeared by letters of Mr. Flamsteed's found after Mr. Sharp's death, many of which I have seen.
" I have been the more particular in what relates to Mr. Sharp, in the business of constructing this mural arc, not only because we may suppose it the first good and valid instrument of the kind, but be cause I look upon Mr Sharp to have been the first person that cut accurate and delicate divisions upon astronomical in struments, of which, independently of Mr Flamsteed's testimony, there still re main considerable proofs; for, after leav ing Mr. Flamsteed, and quitting the de partment above mentioned, he retired into Yorkshire, to the village of Little Horton, near Bradford, where he ended his days about the year 1743 (should be in 174'3), and where I have seen not only a large and very fine collection of me chanical tools, the principal ones being made with his own hands, but also a great variety of scales and instruments made with them, both in wood and brass, the divisions of which were so exquisite, as would not discredit the first artist of the present times; and I believe there is now remaining a quadrant, of four or tire feet radius, framed of wood, but the limb co vered with a brass plate, the subdivi&onis being done by diagonals, the lines of which are as finely cut as those upon the quadrants at Greenwich. The delicacy of Mr. Sharp's hand will indeed perma nently appear from the copper-plate in a quarto book, published in the year 1718, entituled ' Geometry Improved, by A. Sharp, Philomath,' (or rather 1717, by A. S. Philomieh,) whereof not only the geo metrical lines upon the plates, but the whole of the engraving of letters and fi gures were done by himself; as I was told by a person in the mathematical line, who very frequently attended Mr. Sharp in the latter part of his life. I therefore look upon Mr. Sharp as the first person that brought the affair of hand division to any degree of perfection " Mr. Sharp kept up a correspondence by letters with most of the eminent ma thematicians and astronomers of his time, as Mr. Flamsteed, Sir Isaac Newton, Dr. Halley, Dr. Wallis, Mr Hodgson, Mr. Sherwin, &c. the answers to which letters are all written upon the backs or empty spaces of the letters lie received, in a shirt hand of his own contrivance. From a great of letters (of which a large chest-full remains with his friends); from these and many other well known facts it is evident, that Mr. Sharp spared neither pains nor time to promote real science. Indeed. being one of the most accurate and indefatigable computers that ever existed, he was for many years the common resource for Mr. Flamsteed, Sir Jonas Moore, Dr. Halley, and others, in all sorts of troublesome and delicate calculat ions.