F A C E. wont to be called geometry (a word taken from ancient use, because it was first applied only to measuring the earth, and fixing the limits of poffeffions), though the name seemed very ridiculous to Plato, who subítitutes in its place that more extenfive name of Metrics or Mensuration; and others after him gave it the title of Pantometry, because it teaches the method of measuring all kinds of magnitudes." But notwithstanding the dignity and importance of this subject, the books which have lately been offered to the public under the title of mensuration, 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 subject in a more favourable. light, and in some measure to endeavour to retrieve its reputation. The work consists of five principal parts, each part being divided into several sections, every fection containing several problems or propofitions, which are all demonstrated after the fimplest and shortest methods that could be devised; the process being sometimes by pure geometry, and sometimes by algebra, the method of fluxions, the method of increments, &c, according as the one or the other appeared best suited to the purpose. And thus, by uling various modes of demonstration, I was enabled to make this work much more complete than it could otherwise have been; not only by employing that particular method which would perform the butiness in the shorteft or clearest manner, but which also would render the subject still more extensive. However, I had not always my choice of the method of demonstration, the subject fometimes requiring one mode, and sometimes another; for although the inethod of fluxions be generally the most concise, and, in other respects, the most proper for investigating the measure of extensions, yet there are some things in this business which are too fimple for it, as well as some others which rise 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 supposed, and its result assumed in the very notation ; for the rectangle which is considered as the fluxion of a surface, is denoted by its length drawn into its breadth: and on this account those fluxionilts, who, in their chapter of quadratures, attempt to determine the measure of a rectangle, argue in a circle, or deduce as a conclusion what was necessarily supposed in the premises : the same may be said with regard to prisms. 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 base, and of an infinite length, it being there necessary to call in the affiftance of the method of increments, As As extensions are of three kinds, longitudinal, fuperficial, and folid, fo in this work the science is treated of as distinguished by nature into these three principal parts; that is, so far as the nature of geometrical figures would conveniently admit. With regard to right lines, plane surfaces, and solids, the distinction is general; but it does not obtain with regard to curved lines and furfaces. For, by preserving this distinction fo entire, as to determine the measure of all kinds of lines in the first part, and of all kinds of surfaces in the second, 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 second part, only of the measure of planes which are bounded by right and circular lines, without any curve furfaces, excepting that of the sphere. In the third part, which treats of folids, I have generally placed the problems which relate to the measures of the lines, surfaces, and solidities 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 several interesting practical subjects in life. So much for the distribution and order of the parts in general. I shall now proceed to describe the contents of the parts themselves more particularly. The whole work consists of five parts. Part I. Contains the menfuration of right lines and rightangled angles, and is divided into three fcctions. Sect. 1. Contains several geometrical definitions and problems; fome of which are new, and, it is presumed, they are all more complete, and less exceptionable, than those initead of which they have been substituted. The problems will prepare the learner for making the several figures which are afterwards arcated of. Seet. 2. Contains plane Trigonometry, or the measuring of lines and angles. All the cases of trigonometry, both right and oblique angled, are here reduced to three only; by which means they are cater to be remembered, and more clearly underitood. Besides these cases, which perform the business in the common way, by means of the fines, tangents, and secants of angles, I have given a new and extensive method, by which all the cases of trigonometry are performed independent of fines, tangents, and fecants, and without any kind of tables. Seet. 3. Contains the application of trigonometry to the determination of heights and distances ; in which a great variety of cases and methods concerning this curious subject are explained. Part II. Treats of fuperficial menfuration, or the mensuration of plane figures, and is divided into two lections. ST. 1 a xiii Sect. 1. Treats of the areas, &c. of right-lined and circular sigures ; in which, besides many things that are new and curious, are given an explanation of Profeffor Machin's celebrated quadrature of the circle, and the demonstrations of some useful approximations to the measures of circular arcs and areas, which had been given by Mr. Huygens and Sir Isaac Newton, without demonítrations. Se27. 2. Contains a curious and useful collection of questions concerning areas, promiscuously placed, and resolved by the rules in the former sections. PART III. Contains the measuring of solids, and is divided into 8 sections. Sect. 1. Treats of bodies that are bounded by right or circular lines, viz. prisms, pyramids, the sphere, and the circular spindle. Sect. 2. Treats of the five regular solids or bodies. 4. Treats of the conic sections in general ; and though it be short, it contains several things that are new and of great importance. Sect. 5. Treats of the ellipse and the figures generated by it, viz. spheroids and elliptic spindles. Sect. 6. In like manner treats of parabolic lines, areas, surfaces, and solidities. And Sect. 7. Of hyperbolic lines, areas, furfaces, and solidities. In these sections the several 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 several things which cannot be truly expressed 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 promiscuous collection of questions concerning folids, to exercise the learner in the foregoing rules. Part IV. Contains, in 3 sections, several subjects relating to menfuration in general. Sect. 1. Contains a treatise on the true quadrature and cubature of curves in general. In which are contained some of the most universal and important propofitions that can be made in the subject. Sect. 2. Contains the equidistant-ordinate method ; or, the approximate quadrature and cubature of curves in general, by means of equidistant ordinates or sections. A subject by which general and finite rules are discovered for all figures ; for fome of which they are accurately true, and for the others they are very Dear approximations, which are often the most usefui ruies that can be applied to many things in real pracace. $07. Sct. 3. Contains, in a very concise but copious treatise, the relations between the areas and solidities of figures, and the çenters of gravity of their generating lines and planes. Then the Fifth ard last Part, in four sections, contains the application of the general rules to the most useful subjects of measuring that happen in ordinary life. In these subjects very naterial improve. ments are almost every where made, both with respect to the matters and the disposition of them. Sect. 1. Contains a very fimple treatise of land surveying ; explaining the use of the instruments, the methods of surveying, 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 $22. 3. On the measuring of artificers works; viz, Bricklayers, Masons, Carpenters and Joiners, Slaters, and Tilers, Platterers, Painters, Glaziers, Pavers, and Plumbers. Containing the description of the carpenter's rule, the several measures used by each, with the methods of taking the dimensions, and of squaring and suniming them up. The whole illustrated by a real case of a building, in which are thewn the methods of entering the dimensions and contents in the pocket book, of drawing out the abstracts, and from them drawing out the forms of the bills. Seel. 4. Contains a curious treatise on timber measuring ; in which, among several other things, is given a new rule for meafuring round timber, which not only gives the content very exact, but it is at the same time as easy in the operation as the common false one, either by the pen or the fliding rule. It contains also iome curious rules for cutting timber to the most advantage. The book then concludes with a large table of the areas of circular segments, extended to ten times the usual length. It may be necessary to remark that, in this book, where a curve or a space is said to be non-quadrable, or it is said that the value of a thing cannot be expreffed except by an infinite series, or any fuch-like expreffion is used; the meaning is, that it is not gcometrically 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 used; but not that it is a thing naturally impossible in itself. For although a space be not quadrable, by the methods yet known, it does not therefore follow that its quadrature is an impossible thing, or that some method may not hereafter be difcovered by which it may be squared. All the methods used by the geometricians before Archimedes, were insufficient for the quadrature of any curve space whatever ; but were they there. fore to infer that no curve could by any means be squared? Archimedes discovered a method by which he squared the parabola; and by the lately-discovered method of Auxions, we can find as many quadrable curves as we please. It is true we have not yet found the area of the circle, and several other figures, in finite terms ; yet for each of these we can assign infinite series whose laws of progression are visible ; which is more than the ancients could do, or perhaps ever expected could be done, if they even at all thought of such things. · And, perhaps, hereafter a method may be discovered of squaring any figure whatever. Which is the chicf problem in geometry. In this edition have been made many large and useful additions, in almost every section of the work; and it is presumed that the whole is arranged in a more regular and perfect order than before. Royal Military Academy, Jan. 24, 1788. |