saving of space has been effected, enabling the Proprietors to add a complete Treatise on Trigonometry without increasing the size of the book. They have, therefore, great pleasure in introducing the present Edition, believing it to be the cheapest and the best school-book on the subject that has ever been presented to the Public.

In this Edition two new Parts have been introduced. Part the Ninth treats of Plane and Oblique Trigonometry, with a variety of illustrations and practical examples. It will be seen, this subject is treated, upon an entirely new, and improved method, by substituting Arithmetical Complements in the Logarithmic Numbers.

Part the Tenth treats of the Application of Plane Trigonometry to the Mensuration of Heights and Distances in a variety of useful questions; the method of finding the Heights of Mountains by means of the Barometer and Thermometer; also numerous examples for finding the magnitude and dimensions of the earth; the vibration of pendulums in different latitudes, on the length of arcs of different meridians, and a great variety of other important matter.

LONDON, 1844.

PREFACE.

VARIOUS have been the conjectures concerning the origin of Geometry or Mensuration; but as it is a Science of general utility, there can, I think, be little doubt that its existence is nearly coeval with the creation of Man. Indeed I can see no reasonable objection why we may not attribute its invention to our first parent Adam; especially as we are informed in Holy Writ, that his son Cain built a city; to do which, it is evident, would require some knowledge of a measuring unit, which is the first principle of Mensuration. By the same infallible testimony, we find that the Arts and Sciences were cultivated to a considerable extent long before the Flood. Jubal was the father of all such as handled the harp and organ; and Tubal-cain, an instructor of every artificer in brass and iron. It is also more than probable that Noah was well acquainted with the Art of Mensuration, as practised in his day; for it does not appear that he found any difficulty in building the Ark, which consisted of three stories, according to certain dimensions given him by the Lord of the Universe.

Diodorus, Herodotus, and Strabo, are of opinion that the Science of Mensuration had its rise among the Egyptians; whom they represent as constrained, on account of the removal or defacing of the land-marks by the annual inundation of the Nile, to devise some method of ascertaining the ancient boundaries, after the waters had retired. By Josephus, however, the invention is ascribed to the Hebrews. This writer asserts that the Arts and Sciences, for which Egypt was so long famous, were carried into that country, by the Patriarch Abraham, from Ur of the Chaldees; but as Egypt was peopled by the descendants of Ham, is it not more probable that they had the rudiments of their Sciences originally from their father?

Be this as it may, it is well known that Egypt was, for

many ages, the mother and nurse of the Arts and Sciences. From this country they were conveyed into Greece by Thales, about 600 years before the Christian Era. This Philosopher, after travelling into Egypt, and studying, under its sages, Astronomy, Geometry, and other branches of the Mathematics, returned to his own country, and employed himself in communicating the knowledge which he had acquired.

The great utility of Mensuration, without which it is impossible rightly to conduct the affairs of civilized life, induced many of the most celebrated Philosophers and Mathematicians of antiquity to study its principles; and to Thales, Anaxagoras, Pythagoras, Socrates, Plato, Apollonius, Philo, Ptolemy, Aristotle, Euclid, Archimedes, &c. we are indebted for many substantial improvements. The moderns, likewise, have not been less solicitous to enrich this Science, than the ancients; accordingly Huygens, Wallis, Gregory, Halley, Euler, Leibnitz, the Bernouilles, Vieta, Metius, Van Ceulen, Barrow, Newton, Sharp, Machin, Moss, Leadbetter, Simpson, Emerson, Holliday, Fletcher, Robertson, Hutton, Bonnycastle, Keith, Beckett, &c. have greatly improved it by their labours.

After so many eminent men have written more or less upon this Science, it may perhaps be thought presumption in me to attempt to add any thing to its stores; but as I can say, without arrogance, that I have had considerable experience in the Practical Part of Mensuration, in all its Departments, I am persuaded that this work will be found to contain many things both new and useful.

With regard to the Rules, indeed, nothing new can be expected; as they are to be found, with very little variation, in every modern Treatise on Mensuration; but the Questions, which amount to five hundred and eighty, are almost wholly new ones; and a great number of them have been made from actual admeasurements. I have

likewise given copious directions for taking dimensions, which art certainly forms a very essential part of Mensuration; for if the dimensions be improperly taken, the results must, of course, be incorrect.

The work is divided into Ten Parts; and some of these Parts are again subdivided into Sections.

Prat the First contains Practical Geometry, and a few of the most useful Geometrical Theorems; most of which

are employed in solving Questions in this Work. The Theorems are not demonstrated; but references are given to the elements of Euclid, Simpson, and Emerson, where their demonstrations may be found.

Part the Second contains Mensuration of Superficies. In the last Problem of this Part, I have given the invaluable Rule for finding the areas of curvilineal figures, by means of equi-distant ordinates. This Rule was first demonstrated by the illustrious Newton; but it is to Mr. Thomas Moss that we owe its present simplicity.

Part the Third is divided into two Sections. In the first are given the Methods of surveying and planning single Fields, Woods, Roads, and Rivers; and also Rules and Directions for Parting off, and Dividing Land. It likewise exhibits four of the most approved Methods of surveying large Estates, illustrated by three distinct Plans, and Field-Books.

Sometimes one of these Methods claims the preference, and sometimes another, according to the different forms of Estates; but they are all approved of by our best Land Surveyors; and are more accurate and practical than any others, that have come to my knowledge.

This Section will be found to contain every thing that is necessary for persons, in general, to know of Surveying. In order to become a complete Surveyor, it is requisite not only to study Works written professedly on the subject, but also to have a considerable portion of Field-practice, under the direction of an able Tutor.

The second Section contains a collection of useful Questions concerning Superficial Mensuration, which will serve to exercise the ingenuity of the Learner; and to prove his knowledge of the Theorems and Rules given in the first three Parts.

Part the Fourth is divided into four Sections. The first contains the Mensuration of Solids; the second, the Description and Use of the Carpenter's Rule; and the third, Timber Measure.

The last Problem of this Section contains Rules and Directions for measuring and valuing standing Timber; many of which were never before published.

In this part of the Work, I have been assisted by Mr. Joseph Webster, of Farnley, near Leeds; who has been many years very extensively employed, as a valuer of timber, by the Earl of Cardigan, Lord Mexborough, &c. &c.

In order to render this Problem as useful as possible, I have given a description of Timber Trees, and pointed out the purposes for which their wood is best adapted; for it is impossible to become a valuer of timber without being made acquainted with the properties of trees.

The fourth Section contains Miscellaneous Questions concerning the Mensuration of Solids.

Part the Fifth treats of the Method of measuring the Works of Artificers; viz. Bricklayers, Masons, Carpenters and Joiners, Slaters and Tilers, Plasterers, Painters, Glaziers, Plumbers, and Pavers.

The directions in this Part, for taking the dimensions, making the deductions, &c. will be found to be very copious. For some of these I am indebted to Mr. Benjamin Jackson, senior, an able and experienced Architect in Leeds; and to Mr. Joseph Brooke, Teacher of the Mathematics, at Wortley, near Leeds, who has had much experience in measuring the Works of Artificers.

This Part is concluded with a general Illustration, containing the dimensions of a House; and exhibiting the methods of ruling the Book, entering the dimensions with the contents; and forming the bills for workmanship and materials.

Part the Sixth treats of the Mensuration of Haystacks, Drains, Canals, Marlpits, Embankments, Ponds, Milldams, Quarries, Coal-heaps, Clay-heaps, and other irregular figures, by means of equi-distant, parallel sections, founded upon the method of equi-distant ordinates.

This method of finding the contents of irregular figures is pointed out by Dr. Hutton, in his valuable Treatise on Mensuration, octavo*, page 375; but Mr. Joseph Beckett appears to have been the first who has applied it with any degree of success.

This Part also contains the method of measuring the circular Ponds made upon the Wolds in Yorkshire. This was communicated to me by the Rev. W. Putsey, Master of the Classical, Commercial, and Mathematical Academy, Pickering.

In order to give the young Reader an idea of the great improvements made in Agriculture and Commerce, by means of Drains and Canals; and also to make him acquainted with some of the stupendous Works which have been accomplished by the ingenuity, perseverance, and

*Price 18s. in boards.