few unsuccessful trials he will, propably, give up all such applications as useless, or even prejudicial in their results, and become for the rest of his life a believer in the essential and irremediable oppositions of theory and practice. This is not altogether an imaginary history of the fruits of a scientifi education to a practical man. It remains, nevertheless, an incontrovertible fact that there can be no opposition of theory and practice; that not being true theory which in its application to any practical result does not accord with it, nor that sound practice which is opposed to any true deduction of theory. It is in the variety of the circumstances under which the applications of theory are required, and in the complication of those circumstances in each case of such application, that lies the whole difficulty-a difficulty only to be overcome by a long continued and a systematic course of education specially directed to that end-by the constant habit of considering that which is learned in the light of that which is also to be applied, and by continually making this application to such questions as actually present themselves in practice The daily avocations of a dock-yard suggest an abundance of such applications, from the simplest on which the ingenuity of the student may be exercised to the most complicated. And it is impossible to overrate the value, for the purposes of this institution, of that complete mastery over whatever resources of mathematical investigation are placed at the disposal of the students that firmness and certainty of handling them--which might thus be acquired. Impressed with a conviction of the necessity of giving to the studies of the Central Mathematical School from the earliest period of its history this practical bearing, the Principal has, during the past year, directed his attention specially to the science of "Descriptive Geometry,”† a science of which little more than the name is generally known in this country, but which admits of being applied with great advantage to ship-building. Through the medium of this science he has succeeded in giving to the students a knowledge of the methods used to shape the various timbers of a ship so that she may receive the form designed for her by the builder; not as they were heretofore taught, under the form of rules difficult to be remembered because the reasons for them were never understood; but as expedients of a science the principles of which the student knows demonstratively, and which expedients he feels, perhaps, that he could himself, in the application * The question not being invented to suit the theory, but conversely. + Mr. Woolley has written a work on this subject, which is at present in the press, and which cannot but be very valuable to practical men. Technically called "laying off." The rules for laying off the timbers, called diagonals, cant timbers, transoms, stems, &c., and for taking bevellings, all belong to the science of descriptive geometry. of that science, have devised. Whilst the methods hitherto in use cannot under these circumstances but be easier to the student, others will probably present themselves drawn from the same source when, in practice, the occasion for these may arise, and meanwhile the great business of the professional education of his mind thus proceeds under its most legitimate form, and the powers of his understanding are strengthened and developed with a special reference to the circumstances under which he is to be employed in the public service. I have the honor to be, &c. To the Right Honorable the Lords of the HENRY MOSELEY. APPENDIX. Questions proposed to Students of Central Mathematical School in Her Majesty's Dockyard, Portsmouth, Midsummer 1849. (Not more than one question to be answered in each Section.) 1. At what price per head must a farmer purchase a flock of 100 sheep, that expending £10. in feeding them, and losing nine, he may be able to sell the remainder at £2. each and gain £20? 2. I turn over the pages of a book by fours and find three odd ones. I then turn them over by fives and find two odd ones. The last time I do not turn them over so often by twenty times as I did the first. How many pages were there? 3. A passenger train and a luggage train, the one travelling at 10 mile per hour less speed than the other, set out at the same time, the one from London and the other from Carlisle, 210 miles apart, and pass one another at a certain station on the road. The passenger train sets out from Carlisle to return, two hours after the luggage train sets out to return from London; and it is observed that they pass one another at the same station. At what rate do they travel, and how far from London is the station? Section 7. 1. The first term of an arithmetical progression is - 7 the number of terms 8 and the sum 28. What is the common difference? 2. A person sowed a bushel of wheat and the next year he sowed again the whole produce of that bushel, and so on until at the end of the third year he had a bushels. How many grains of wheat must each grain of seed have yielded, supposing it to have yielded the same number every year? 3. What is the present value of an Annuity of £a which increases in geometrical progression whose ratio is r for n years' interest being assumed at p per cent. per annum? Section 8. 1. Approximate by the method of continued fractions to the value of 587 1943 2. Show in the above example that the approximating fractions must be alternately greater and less than the true value. 3. How can the fraction nominators are 7 and 11. 230 be divided into two others whose de. Section 9. 1. Find expressions for the number of years on which £ a will amount to £ A. at r per cent. simple interest, and at r per cent. compound interest. 2. Investigate a general method for expressing a number N in the scale of Notation, whose radix is r: and show that the number, when so expressed, will leave the same remainder when divided by (r − 1), 13 the sum of its digits will. 3. Show, that if one solution of the indeterminate equation a x + by = C be given, all the rest may be determined from it. 1. Expand 2x + 3 3x + 2 Section 10. to four terms in a series ascending by powers of by the method of indeterminate co-efficients. 2. Resolve 3x + 2 3. Expand as in a series ascending by powers of t. Section 11. 1. Investigate an expression for the number of permutations of n things taken r together, and find the number of signals which may be made with six different flags placed one above the other. 2. Prove the Binomial Theorem in the case in which the index is a positive integer. 3. Investigate the general term of the Multinomial Theorem in the case in which the index is a positive integer. Section 12. 1. Show that every number, which is a perfect square, is of one of the forms 5 m or 5 m + 1. 2. Find the number of balls in a triangular pile, each side of the base of which contains 25 balls. 3. The plate of a looking-glass costs £a per square foot, and the frame £b per linear foot of the top and bottom, and £ c per linear foot of the sides. What is the largest glass that can be bought for £ m. Section 13. 1. Show that the perpendicular from the focus of a parabola upon any tangent, intersects that tangent in the same point in which a tangent to the vertex intersects it. 2. Show that the sum of the distances of any point in an ellipse from the foci, is equal to the axis major. 3. Show that the semi-axis major of an ellipse in a mean proportional between the distance, C N, from the centre to the foot of the ordinate to any point, and the distance C T from the centre to the intersection of the tangent, to that point with the axis. Section 14. 1. Investigate the general equation to a straight line, and draw the line whose equation is 2. Find the perpendicular distance of a point whose co-ordinates are xy, from a line whose equation is ax + by = C. 3, Having given the co-ordinates of a front with reference to any given axis, to determine them with reference to an axis inclined to the former at any given angle, the origin being the same. Section 15. 1. Determine the locus represented by the equation, x2 + y2-2 cx + 6cy + 9 c2 = 0. 2. Determine the equation to the tangent to an ellipse. 3. Show that the second term in the general equation of the second degree may be made to disappear by turning the axis of the co-ordinates through a certain angle. Section 16. 1. To inscribe a square in a given triangle, one side being on the side of the triangle. 2. To find the locus of the vertex of a triangle, whose base and the ratio of its sides are given. 3. To find the locus of the middle points of all the chords passing through a given point in an ellipse. VOL. II. 3 M Section 17. 1. Show that the pressures acting on the arms of a lever will balance when they are reciprocally proportional to their distances from the fulcrum. 2. If a point be kept at rest by three pressures, any two are to one ans ther inversely as the sines of the angles they make with the third. 3. To find the centre of gravity of any number of heavy bodies, whose respective centres of gravity do not all lie in the same plane. Section 18. 1. Investigate the relation between the power and the weight in a single moveable pulley, when the strings are not parallel. 2. Show that the centre of gravity of a triangle is situated in the lin joining one of its angles with the bisection of the opposite side at twe thirds the length of that line. 3. Show that a couple will produce the same statical effect upon a rigid body, when it is moved parallel to itself in its own place. 4. Investigate a relation between the space and time when the motion of a body is uniformly accelerated. Section 19. 1. Describe the common steelyard, and give a formula for graduating i 2. A body is projected vertically upwards from the top of a precipier with a velocity of 160 feet per second, and five seconds afterwards a body is let fall from the same place. Both bodies reach the foot of the precipice together. What is its height? 3. One of the cannons of a ship is made to describe a circle on the deck. Show that the centre of gravity of the ship will thereby be made to describe a circle, and compare the diameters of the two circles. 4. At what angle must the double wedges which support the keel of a ship in dock be cut, that its weight may just cause them to slide apart, the co-efficient of friction being supposed to be given. Section 20. 1. Explain the formation of artesian wells. 2. Determine the pressure of a fluid upon a portion of a plane immersed in it, and inclined at any angle to the surface. 3. Show that if a body be not at rest in a fluid, the moving force by which it ascends or descends is the difference between the weight of the solid and that of an equal bulk of the fluid. Section 21. 1. Describe the common hydrometer, and show how it may be graduated. 2. The weight of a bottle full of water is found to be w; a body, whose weight is w, having afterwards been placed in it, the weight of the whole is found to be w1. What is the specific gravity of the body? 3. What are the laws in respect to elastic fluids known as those d Marriotte and Gay-Lussac? Express by a formula the relation betwe the elastic force, density, and temperature of a gas. Section 22. In what way is it c What are its combin 1. What is the chemical constitution of the air? nected with the support of animal and vegetable life? 2. In what different forms does carbon exist? tions with oxygen? How may they be obtained? 3. Explain what is meant by the law of multiple proportions in chemistry. |