Amongst the new subjects which have been introduced in this work may be mentioned, The finding of the metacentre; The displacement of various solids; The areas and centres of gravity of curves by Simpson's rule; The displacement and centre of gravity of a ship, together with its statical and dynamical stability; The length of the helix of a screw-propeller, with its area and angle, by means of a table which will facilitate considerably the computation of these elements. I trust that the whole of this additional matter of increasing interest and importance will be found useful to those who are engaged in the construction and management of the great bulwarks of our country.

The examples, which, in the ordinary rules, are numerous, have been considered with the object of eliciting several answers from the same data, a course which will be best illustrated by means of a quotation, viz., "Find the surface, volume, weight, and displacement of a globe of English oak whose diameter is 24 feet 7 inches." The reasons for such a procedure must be obvious to all engaged in the arduous and important office of teaching. With a view to impress the algebraical formula, which state the rules, more strongly upon the memory, and at the same time to afford considerable practice in algebraical equations, there is given a great number of inverse questions and miscellaneous examples, which require for their solution some acquaintance with the principles and practice of Algebra. In connection with my own experience I have found the working of these inverse questions attended with advantage, both as an exercise in the use of algebraical symbols, and in the solution of equations which commonly cccur in the consideration of practical questions. It has been deemed advisable to follow, generally, the usual plan of stating the rules in ordinary language as well as by algebraical formulæ. I hope, however, as the advantages of education become more extensive and better appreciated by the artisans of this country, that the former mode will be entirely abolished and yield its position to the supremacy of the latter, which is far more intelligible and effective. An acquaintance with the ordinary rules of addition, subtraction, multiplication, and division of Algebra, together with the simplest axioms of Geometry, a task by no means difficult to be accomplished even by the humblest artisan, will readily enable a person to multiply his knowledge manifold with certainty and success. In order to illustrate what I mean I e use of the following relation :

which expresses the weight (W) of a sphere whose diameter is D and density 8. Suppose, in the first place, that the weight and density of the sphere were known quantities and the diameter was required. In this case, both sides of the above equation must be multiplied by 6 and divided by π 8, and the cube root of the result extracted; then the formula will stand thus :

Suppose, in the second place, that the weight and diameter of the sphere were known quantities, and the density was required. Then, both sides of the first equation must be multiplied by 6, and divided by the product of T and D3, and the equation will become

(The density of a material body is the weight of a cubic foot of it.)

The two latter problems are inverse problems, and require for their solution the application of the simplest axioms and the easiest principles. The demonstrations of the rules by the application of geometrical and algebraical formulæ have been, for the most part, omitted, in accordance with the plan which, after considerable experience, has been thought advisable to follow. In the mensuration of lines, areas, and volumes or solids, it is customary for mathematicians to conceive a line to be generated by the motion of a point; an area to be produced by the motion of a line; and a solid or volume to be the result of the motion of a plane; and in order that this motion may be the interpretation of known curved lines, areas, and volumes of various forms, it is necessary to suppose a point in motion to be constantly changing its direction, a line in motion to be constantly changing its length and direction, and a plane in motion to be constantly changing its area and direction.

From this view of the subject it will be obvious, after a little reflection, that, with few exceptions, the artifices and resources of the higher mathematics (particularly the integral calculus) are absolutely necessary to enable any one to investigate, with

success, the rules and formula for the length of curves, areas, and volumes. Feeling the force of this consideration, I could not resist the conviction that the demonstrations of the rules in a work of this kind would be useless to those who are familiar with the principles and processes of the integral calculus, and could not be given in a way to be useful to those unacquainted with the science. With respect to the history of this portion of practical geometry, it would be difficult to record, with accuracy and fidelity, the important labours of each mathematician, whose genius and investigations have contributed so largely to its present advanced state of usefulness; still, the particular history of the famous ancient problem, "Squaring the Circle," is so interesting and well defined as to justify its insertion in a work of this nature. The squaring of a circle is a problem, the object of which is to construct, geometrically, with a rule and compass, a square equal in area to a given circle. The artifices, modes of thought, and profound geometrical skill, which have been employed by the humblest as well as by the most successful cultivators of pure science, in every age and in every civilised country, since the time of Archimedes, have not accomplished the object proposed; but have simply pointed out, to those who can seize the force of their developments, the impossibility of the problem. It may be stated that the " Squaring of the Circle" is readily reduced to the problem, "to find the ratio of the circumference of a circle to its diameter," and it is this enquiry which has received the greatest share of attention. In consequence of the frequent application of the ratio here alluded to, in various branches of mathematics, it is usual to designate it by the symbol π.

Archimedes, the prince of ancient philosophers, was the first mathematician whose labours in connection with this problem were crowned with practical success. He showed that the value of was nearly equal to 22 =3.1428571. This approximation being a little too great is very near the truth, and, in consequence of its simplicity, is still used by many practical men.

7

It was discovered by means of a circumscribed regular polygon of 192 sides, and an inscribed regular polygon of 96 sides.

355
113

Vieta and Metius found the value of to be = 3.1415929,

which is more correct than the ratio of Archimedes, but not sa simple.

By the continual bisection of an arc of a circle, Van Ceulen, a Dutchman, determined the ratio π to be 3.1415926535, &c., to 36 places of decimals. This result was obtained by troublesome and laborious calculations, and was thought so curious that the numbers were cut on his tombstone in St. Peter's Churchyard, at Leyden. Abraham Sharp, of Little Horton, near Bradford, Yorkshire, extended the approximation of Van Ceulen to 72 places of decimals. The discovery of the differential and integral calculus, the summation of infinite series, and the powerful notation of trigonometrical functions, have enabled modern mathematicians to devise series and formulæ by which the value of π can be more: readily approximated to than by the inscribed and circumscribed regular polygons of the ancients.

Machin, Professor of Astronomy in Gresham College, extended the ratio π to 100 places of decimals by meams of the formula,

Before 1831, by the labours of De Lagny, Euler, and other celebrated mathematicians, the value of had been calculated to the extent of 140 places of decimals.

In 1841, Dr. Rutherford, Royal Military Academy, Woolwich, calculated the value of π to 208 decimals, by the formula

T

In 1846, M. Dase found the value of π to 200 decimals from the

In 1847, Dr. Clausen, of Dorpat, pushed the approximation of π to 250 decimals by means of the formulæ

(See "Rectification of the Circle to 607 Places of Decimals," by

Mr. Shanks.)

In 1851, William Shanks, Houghton-le-Spring, Durham, calculated, by means of Machin's formula, the value of to 315 decimals. This gentleman and Dr. Rutherford have since pushed the approximation of to the extent of 441 decimals. The computations were independent of each other, and therefore the decimals to this extent may be pronounced free from errors.

With great industry, Mr. Shanks has pushed the approximation still further, viz., to 607 places of decimals; for the accuracy of the last 166 figures he is alone responsible.

The value of ", as calculated by Mr. Shanks, is 3.1415926 5358979 3238462 6433832 7950288 4197169 3993751 0582097 4944592 3078164 0628620 8998628 0348253 4211706 7982148 0865132 8230664 7093844 6095505 8223172 5359408 1284811 1745028 4102701 9385211 0555964 4622948 9549303 8196442 8810975 6659334 4612847 5648233 7867831 6527120 1909145 6485669 2346034 8610454 3266482 1339360 7260249 1412737 2458700 6606315 5881748 8152092 0962829 2540917 1536436 7892590 3600113 3053054 8820466 5213841 4695194 1511609 4330572 7036575 9591953 0921861 1738193 2611793 1051185 4807446 2379834 7495673 5188575 2724891 2279381 8301194 9129833 6733624 4193664 3086021 3950160 9244807 7230943 6285530 9662027 5569397 9869502 2247499 6206074 9703041 2366929

13332+ &c.