A treatise on mensuration, both in theory and practice. The second edition, with many additions

Sect. 1. Treats of the areas, &c. of right-lined and circular figures; in which, befides many things that are new and curious, are given an explanation of Profeffor Machin's celebrated qua drature of the circle, and the demonftrations of fome useful approximations to the measures of circular arcs and areas, which had been given by Mr. Huygens and Sir Ifaac Newton, without demonftrations.

Sect. 2. Contains a curious and ufeful collection of questions concerning areas, promifcuoufly placed, and refolved by the rules in the former fections.

PART III. Contains the measuring of folids, and is divided into 8 fections.

Sect. 1. Treats of bodies that are bounded by right or circular lines, viz. prisins, pyramids, the sphere, and the circular spindle. Sect. 2. Treats of the five regular folids or bodies.

Scct. 3. Treats of folid rings.

Sect. 4. Treats of the conic fections in general; and though it be fhort, it contains feveral things that are new and of great importance.

Sect. 5. Treats of the ellipfe and the figures generated by it, viz. fpheroids and elliptic fpindles.

Sect. 6. In like manner treats of parabolic lines, areas, surfaces, and folidities. And

Sect. 7. Of hyperbolic lines, areas, furfaces, and folidities.

In these fections the feveral figures and bodies are very extenfively and particularly handled, many of the rules, &c. both here and throughout the whole book, being new and interesting; and I have given throughout many neat approximations to the values of feveral things which cannot be truly expreffed otherwise than by an infinite feries; which approximations are mostly new, excepting two or three that were given by Sir I. Newton, and which I have demonstrated here for the first time.

Sect. 8. Or the last of this part, contains a promifcuous collection of questions concerning folids, to exercife the learner in the foregoing rules.

PART IV. Contains, in 3 fections, feveral fubjects relating to menfuration in general.

Sect. 1. Contains a treatise on the true quadrature and cubature of curves in general. In which are contained fome of the most univerfal and important propositions that can be made in the fubject.

Sect. 2. Contains the equidiftant-ordinate method; or, the approximate quadrature and cubature of curves in general, by means of equidiftant ordinates or fections. A fubject by which general and finite rules are discovered for all figures; for fome of which they are accurateiv true, and for the others they are very near approximations which are often the most ufefui ruies that can be applied to many things in real practice.

Seat.

Sect. 3. Contains, in a very concife but copious treatife, the relations between the areas and folidities of figures, and the centers of gravity of their generating lines and planes.

Then the

Fifth and laft PART, in four fections, contains the application of the general rules to the most useful fubjects of measuring that happen in ordinary life. In thefe fubjects very material improve ments are almost every where made, both with respect to the matters and the difpofition of them.

Sect. 1. Contains a very fimple treatife of land furveying ; explaining the ufe of the inftruments, the methods of furveying, of planning, of computing the contents, of reducing plans, and of dividing the ground.

Sect. 2. Contains a very curious and complete treatise on gauging. As in like manner doth

Sect. 3. On the meafuring of artificers works; viz, Bricklayers, Mafons, Carpenters and Joiners, Slaters, and Tilers, Plasterers, Painters, Glaziers, Pavers, and Plumbers. Containing the defcription of the carpenter's rule, the feveral measures ufed by each, with the methods of taking the dimenfions, and of fquaring and funming them up. The whole illuftrated by a real cafe of a building, in which are fhewn the methods of entering the, dimenfions and contents in the pocket book, of drawing out the abftracts, and from them drawing out the forms of the bills.

Sect. 4. Contains a curious treatife on timber measuring; in which, among feveral other things, is given a new rule for meafuring round timber, which not only gives the content very exact, but it is at the fame time as eafy in the operation as the common falie one, either by the pen or the fliding rule. It contains alfo fome curious rules for cutting timber to the most advantage. The book then concludes with a large table of the areas of circular fegments, extended to ten times the ufual length.

It may be neceffary to remark that, in this book, where a curve or a space is faid to be non-quadrable, or it is faid that the value of a thing cannot be expreffed except by an infinite feries, or any fuch-like expreffion is ufed; the meaning is, that it is not geometrically quadrable, or that its area or value cannot be expreffed in a finite number of terms, by any method yet known, or by the method there ufed; but not that it is a thing naturally impoffible in itfelf. For although a space be not quadrable, by the methods yet known, it does not therefore follow that its quadrature is an impoffible thing, or that fome method may not hereafter be difcovered by which it may be fquared. All the methods used by the geometricians before Archimedes, were infufficient for the quadrature of any curve space whatever; but were they therefore to infer that no curve could by any means be fquared? Archimedes difcovered a method by which he fquared the parabola; and by the lately-difcovered method of fluxions, we can

find as many quadrable curves as we please. It is true we have
not yet found the area of the circle, and feveral other figures, in
finite terms; yet for each of these we can affign infinite feries
whofe laws of progreffion are vifible; which is more than the
ancients could do, or perhaps ever expected could be done, if they
even at all thought of fuch things. And, perhaps, hereafter a
method may be difcovered of fquaring any figure whatever.
Which is the chief problem in geometry.

In this edition have been made many large and useful additions,
in almost every fection of the work; and it is prefumed that the
whole is arranged in a more regular and perfect order than before.

Royal Military Academy,

Jan. 24, 1788.

easy Rule to find the Cube Root with an exacple

what

say is the crabe root of 12812 904 Rivide the given number into periods of I figures, begining at the place of crits, place the cube root of the per 12 perios 2. fur the devotions & subtract it's cube 8 from the in first period & bring down the next periti for a dividend which is 11812 to find a divisor, thuttiply the square of the figure places in the quotient by 300 = 1200 find how oftens this is contained in the divion viz. & times, place the & in the quotient for the 200 figur the rook, Multiply the part of the Rook fomerly found 3.2 by the last figure placed in the book, wings, to the prod uct by 30=180 and thus & the square of the last figure placed in the root to the divisor viz. 1200 Multiple, the sun "therer 1387 by the last figure placed in the root, &, auto sus tract the prosect 4/1 67 from the discord 4812, bring down another period for. a new dividend & procid in the same