wont to be called geometry (a word taken from ancient use, because it was firft applied only to measuring the earth, and fixing the limits of poffeffions), though the name feemed very ridiculous to Plato, who fubítitutes in its place that more extenfive name of Metrics or Menfuration; and others after him gave it the title of Pantometry, becaufe it teaches the method of measuring all kinds of magnitudes."

But notwithstanding the dignity and importance of this fubject, the books which have lately been offered to the public under the title of menfuration, have treated it in a very unworthy manner, and have brought it into much contempt. It is intended in this work, therefore, to place the fubject in a more favourable light, and in fome measure to endeavour to retrieve its reputation.

The work confifts of five principal parts, each part being divided into feveral fections, every fection containing feveral problems or propofitions, which are all demonftrated after the fimpleft and fhorteft methods that could be devifed; the procefs being fometimes by pure geometry, and fometimes by algebra, the method of fluxions, the method of increments, &c, according as the one or the other appeared best fuited to the purpose. And thus, by uling various modes of demonstration, I was enabled to make this work much more complete than it could otherwife have been; not only by employing that particular method which would perform the bufinefs in the fhorteft or clearest manner, but which also would render the fubject ftill more extenfive. However, I had not always my choice of the method of demonftration, the fubject fometimes requiring one mode, and fometimes another; for although the method of fluxions be generally the most concife, and, in other respects, the moft proper for inveftigating the measure of extenfions, yet there are fome things in this bufinefs which are too fimple for it, as well as fome others which rife above its reach: thus, the method of fluxions cannot determine the measure of a rectangle, that determination being below its root, being prior to the fluxionary method of determining areas; for in this mode, that determination is fuppofed, and its refult affumed in the very notation; for the rectangle which is confidered as the fluxion of a furface, is denoted by its length drawn into its breadth: and on this account thofe fluxionifts, who, in their chapter of quadratures, attempt to determine the measure of a rectangle, argue in a circle, or deduce as a conclufion what was neceffarily fuppofed in the premises: the fame may be faid with regard to prifms. And, on the other hand, that method alone would not extend to the determination of the equality of a triangle and hyperbola of the fame bafe, and of an infinite length, it being there neceffary to call in the affiftance of the method of increments.

As

As extenfions are of three kinds, longitudinal, fuperficial, and folid, fo in this work the fcience is treated of as diftinguished by nature into these three principal parts; that is, fo far as the nature of geometrical figures would conveniently admit. With regard to right lines, plane furfaces, and folids, the distinction. is general; but it does not obtain with regard to curved lines and furfaces. For, by preferving this diftinction fo entire, as to determine the measure of all kinds of lines in the first part, and of all kinds of furfaces in the fecond, I immediately perceived that the book could not be fo conveniently adapted to the proper ftudy of the generality of readers: on which account, in the first part, I have treated only of the measure of right lines; and in the fecond part, only of the measure of planes which are bounded by right and circular lines, without any curve furfaces, excepting that of the fphere. In the third part, which treats of folids, I have generally placed the problems which relate to the measures of the lines, furfaces, and folidities of each particular figure, immediately after each other, because the knowledge of the one commonly led to that of the others. The latter parts of the book are employed chiefly about the applications of the general problems to feveral interefting practical subjects in life.

So much for the diftribution and order of the parts in general. I fhall now proceed to defcribe the contents of the parts themselves more particularly.

The whole work confifts of five parts.

PART I. Contains the menfuration of right lines and rightangled angles, and is divided into three fections.

Sect. 1. Contains feveral geometrical definitions and problems; fome of which are new, and, it is prefumed, they are all more complete, and lefs exceptionable, than thofe instead of which they have been fubftituted. The problems will prepare the learner for making the feveral figures which are afterwards areated of.

Sect. 2. Contains plane Trigonometry, or the measuring of lines and angles. All the cafes of trigonometry, both right and oblique angled, are here reduced to three only; by which means they are catier to be remembered, and more clearly underitood. Befides thefe cafes, which perform the business in the common way, by means of the fines, tangents, and fecauts of angles, I have given a new and extenfive method, by which all the cafes of trigonometry are performed independent of fines, tangents, and fecants, and without any kind of tables.

Sect. 3. Contains the application of trigonometry to the determination of heights and diftances; in which a great variety of cafes and methods concerning this curious fubject are explained. PART II. Treats of fuperficial menfuration, or the menfuration of plane figures, and is divided into two fections.

Sect

Sec. 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 quadrature of the circle, and the demonftrations of fome ufeful 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, promifcuously placed, and refolved by the rules in the former fections.

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

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

Sect. 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, furfaces, and folidities.

And

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

In thefe fections the feveral figures and bodies are very exten fively 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 treatife on the true quadrature and cubature of curves in general. In which are contained fome of the most univerfal and important propofitions 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 difcovered for all figures; for fome of which they are accurately true, and for the others they are very near approximations which are often the most ufefui ruics that san be applied to many things in real practice.

Sea.

Sect. 3. Contains, in a very concife but copious treatise, 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 moft ufeful fubjects of measuring that happen in ordinary life. In thefe fubjects very material improvements are almost every where made, both with refpect to the matters and the difpofition of them.

Sect. 1. Contains a very fimple treatise of land furveying; explaining the use 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 measuring 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 fumming 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 abstracts, 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 falfe 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 fpace 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 thefe we can affign infinite feries whofe laws of progreffion are visible; 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 ufeful additions, in almost every section 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.