screw-work from the hand-divisions, as had appeared before in the work of Mr. Tompion: and hence we may conclude, that the method of Dr. Hooke, being executed by two such masterly hands as Tompion and Sharp, and found defective, is, in reality, not to be depended upon in nice mat

ters.

"From the account of Mr. Flamsteed, it appears also, that Mr. Sharp obtained the zenith point of the instrument, or line of collimation, by observation of the zenith stars, with the face of the instrument on the 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 balancing weights, to relieve the hand that was to move it of a great part of its gravity. Mr. Sharp continued in strict correspondence with Mr. Flamsteed as long as he lived, as appears by letters of Mr. Flamsteed's, found after Mr. Sharp's death, many of which I have

seen.

66

I have been the more particular relating 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 because I look upon Mr. Sharp to have been the first person that cut accurate and delicate divisions upon astronomical instruments; of which, independent of Mr. Flamsteed's testimony, there still remains considerable proofs: for after leaving Mr. Flamsteed and quitting the department above-mentioned, he retired to the village of Little-Horton, near Bradford, Yorkshire, where he ended his days about the year 1742 and where I have seen not only a large and very fine collection of mechanical 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 are so exquisite, as would not discredit the first artists of the present times: and I believe there is now remaining a quadrant of 4 or 5 feet radius, framed of wood, but the limb covered with a brass-plate; the subdivisions 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 permanently appear from the copper-plates in a quarto book published in the year 1718, intitled, "Geometry Improved, by A. Sharp, Philomath.; whereof not only the geometrical lines upon the plates, but the whole of the engraving of letters and figures, 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."

After such a testimony from the first civil engineer in the kingdom, and the constructor of the Eddystone Light-House, who was also himself originally a matheinatical instrument-maker, we have no occasion to seek for farther proofs of his skill as a mechanist, and any thing which we

• Mr. Smeaton has fallen into a slight error, arising probably from his quoting from memory. The book was entitled, "Geometry Improved, by A. S. Philomath." and pubJished in 1717.

could say would be rather derogatory from his well-earned fame than otherwise.

He continued to assist Mr. Flamsteed in his observations on the meridional zenith distances of the fixed stars, sun, moon, and planets, with the time of their transits over the meridian, the diameter of the sun and moon, and their eclipses, with those of Jupiter's satellites; the variation of the compass, &c. He also assisted him in making his catalogue of 3000 stars as to their longitudes and magnitudes, their right ascensions and polar distances, with the variations of the same whilst they change their longitude. Mr. Sharp also gave very considerable assistance in the calculation of the tables in the second volume of the Historia Cœlestis. He drew charts of all the constellations visible in our hemisphere; and still more beautifully, planispheres of the northern and southern constellations. As the English engravers were at that time considered inferior to those of the continent, these drawings were sent to Amsterdam with a view to have justice done to the ingenuity of our author by their publication; but the originals far surpassed the engravings both in beauty and accuracy. The drawings and proofs of the engravings are preserved with due care and respect by his friends at Horton.

That elaborate and much-admired work referred to by Mr. Smeaton contains a proof of the great extent of his mathematical knowledge, and his persevering labour in calculation. The first part contains a large and ac curate table of circular segments, 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 table exemplified in making logarithms, or their natural numbers to 60 places of figures; and a table of them for all numbers to 100 and primes to 1100, true to 61 places. These are republished in Dr. Hutton's very excellent collection of mathematical tables (Tab. V. p. 203. 5th ed. 1811,) being the logarithms of 254 numbers. The second part is a very neat tract on Polyedra, or solid bodies of many bases, regular and irregular: to which are aded twelve new ones, with the various methods of forming them, and their exact dimensions in surds or species, and in numbers, illustrated by several copper-plates, neatly engraved by himself. He also cut models of these polyedra very neatly in box-wood.

He made many other instruments besides those mentioned by Mr. Smeaton, which are preserved, as well as some which are undoubtedly lost. Amongst these were a curious armillary sphere, which, besides the common properties, has moveable circles and other apparatus for the exhibition and resolution of spherical triangles, a double sector, sextants, quadrants, and dials of various kinds; all contrived, graduated, and finished by himself in the most elegant manner. These instruments were as curious as they are valuable, and beautiful as they are accurate. Indeed so very exact was he in his graduations, that none of the mathematical instru ment-makers could surpass him at any time, and but very seldom equal

him.

It is not certain how long he stayed at Greenwich, but we find he stayed long enough to greatly impair his health by exposure to the chilling damps

of midnight, and the respiration of its unwholesome vapours. He therefore obtained leave to retire, and settled at Horton, where the influence of his native air, and the kind attentions of his friends, soon restored him to a moderate degree of health. Here he began to fit up an observatory of his own, having first made an engine for turning all kinds of work in wood or brass, with a maundrel for turning irregular figures, as ovals, roses, wreathed pillars, &c. He also made himself most of the tools used by opticians, clock-makers, joiners, mathematical instrument-makers, &c.

The limbs or arcs of his large equatorial instrument he divided with the nicest accuracy into degrees and minutes. He used no telescopes but what he made himself, the lenses of which were ground, figured, and adjusted with his own bands.

In the year 1699 he undertook for his own amusement the quadrature of the circle, which he carried to 72 places of figures, 3:1415926535,89793. 23846,2643383279,5028841971,6939937510,5820974944,5923078164, 06, being twice the number which Van Ceulen deduced. He verified it by another calculation from a different series.

He also calculated the logarithmic sines, tangents, and secants, to every second in the first minute of the quadrant; the laborious investiga. tion of which may be seen in the archives of the Royal Society, Mr. Pa trick Murdoch having presented them for that purpose. It contains a proof of the admirable beauty with which he wrote down all his calculations, and, perhaps, is superior to what any penman now living can perform.

He kept up a correspondence with the most eminent men in mathematical science who lived in his time, amongst whom were Flamsteed, Newton, Halley, Wallis, Hodgson, and Sherwin, copies of the answers to whose letters were generally written upon the backs or empty spaces of them, in a short-hand of his own invention. He was the common resource of several of them in all troublesome and delicate calculations.

After he retired to his native village, he fitted up his house with the utmost degree of regularity and neatness, appropriating different apartments for purposes connected with his pursuits, into four or five of which none of the family were permitted to enter without his express permission. He kept very little company, as indeed might be expected; a country village. not being the place for an abstruse philosopher to meet with men of a congenial disposition. He was, however, sometimes visited by two gentlemen of Bradford, the one a mathematician, and the other a medical man; the signal of their arrival was the rubbing a brick against a certain part of the outside wall of the house, which if he did not answer he was not to be seen.

He was a member of the dissenting congregation at Bradford, and was constant in his attendance every Sunday, at which time he took care to be provided with plenty of halfpence; these he very charitably suffered to be taken singly out of his hand, which he held behind him for that purpose, by a number of poor people who followed him in his road to the chapel, without ever turning to see who they were, or asking a single question. He was very spare in his diet, and extremely irregular in his meals. In

order to his not being disturbed, he contrived a little cupboard, something like a window, with a door at the back and a slide in the front, so placed that his servant could, from an adjoining room, put his victuals into the box without any noise, and when he found time and inclination he resorted to his store; but it has often happened that three or four meals have been standing there together, untouched, the servant having taken each at the proper time. It is a fact that cavities were worn in an old English oak table, upon which he used to write, by the frequent rubbing and wearing of his elbows. He continued a bachelor during his whole life; and, perhaps, it may be said of him, as it was of Sir Isaac Newton, "that he never had any time to spare for love." His stature was about the middle size; he was always thin and of a weak constitution, and for the last few years of his life was very feeble. He died July 18th, 1742, in the 91st year of his age.

We scarcely know whether to admire most his extraordinary perseverance, or his versatility as a mechanic. There is, certainly, something very uncommon in his going from the smithy to the graver or the pen, and thence to his calculations, and acquiring a proficiency in each superior to others who had made only one of them their study: but great men are wonders in every age.

Mathematical Repository.

D.

AN ESSAY ON THE USEFULNESS OF MATHEMATICAL LEARNING.

[Continued from page 104.]

THE second advantage which the mind reaps from mathematical knowledge, is a habit of clear, demonstrative, and methodical reasoning. We are contrived by nature to learn by imitation more than by precept, and I believe in that respect, reasoning is much like other inferior arts, (as dancing, writing, &c.) acquired by practice. By accustoming ourselves to rea son closely about quantity, we acquire a habit of doing so in other things. It is surprising to see what superficial, inconsequential reasonings satisfy the greatest part of mankind. A piece of wit, a jest, a simile, a quotation of an author, passes for a mighty argument: with such things as these are the most parts of authors stuffed: and from these weighty premises they infer their conclusions. This weakness and effeminacy of mankind, in being persuaded where they are delighted, have made them the sport of orators, poets, and men of wit. Those lumina orationis are indeed very good diversion for the fancy, but are not the proper business of the understanding; and where a man pretends to write on abstract subjects in a scientifical method, he ought not to debauch in them. Logical precepts are more useful; nay, they are absolutely necessary for a rule of formal arguing in public disputations, and confounding an obstinate and per verse adversary, and exposing him to the audience or readers. But in the

*

search of truth, an imitation of the method of the geometers will carry a man further that all the dialectical rules. Their analysis is the proper model we ought to form ourselves upon, and imitate in the regular disposition and gradual progress of our enquiries; and even he who is ignorant of the nature of mathematical analysis uses a method somewhat analogous to it. The composition of the geometers on their method of demonstrating truth, already found out, viz. by definitions of words agreed upon, by self-evident truths, and propositions that have been already demonstrated, is practicable in other subjects, though not to the same perfection; the natural want of evidence in the things themselves not allowing it; but it is imitable to a considerable degree. I dare appeal to some writings of our own age and nation, the authors of which have been mathematically inclined. I shall add no more on this head, but that one who is accustomed to the methodical systems of truths, which the geometers have reared up in the several branches of those sciences which they have cultivated, will hardly bear with the confusion and disorder of other sciences, but endeavour, as far as he can, to reform them.

Thirdly, mathematical knowledge adds a manly vigour to the mind, frees it from prejudice, credulity, and superstition. This it does two ways, 1st. by accustoming us to examine, and not to take things upon trust: 2ndly. by giving us a clear and extensive knowledge of the system of the world, which, as it creates in us the most profound reverence of the almighty and wise Creator so it frees us from the mean and narrow thoughts, which ignorance and superstition are apt to beget.

The mathematics are friends to religion, inasmuch as they charm the passions, restrain the impetuosity of imagination, and purge the mind from error and prejudice. Vice is error, confusion, and false reasoning. end all truth is more or less opposite to it. Besides, mathematical studies may serve for a pleasant entertainment for those hours, which young men are apt to throw away upon their vices; the delightfulness of them being such, as to make solitude not only easy but desirable.

What I have said may serve to recommend mathematics for acquiring a vigorous constitution of mind; for which purpose they are as useful, as exercise is for procuring health and strength to the body.

I proceed now to show their vast extent and usefulness in other parts of knowledge. And here it might suffice to tell you, that mathematics is the science of quantity, or the art of reasoning about things that are capable of more and less, and that the most part of the objects of our knowledge are such; as matter, space, number, time, motion, gravity, &c. We have but imperfect ideas of things without quantity, and as imperfect a one of quantity itself, without the help of mathematics. All the visible works of God Almighty are made in number, weight, and measure; therefore to consider them, we ought to understand arithmetic, geometry, and statics; and the greater advances we make in those arts, the more capable we are of considering such things as are the ordinary objects of our conception. But this will further appear from particulars.

[To be continued.]