To those students who, apart from their preparation for the First B.A. Examination, wish to pursue the subject in its practical and mercantile applications we would favourably introduce Professor Elliott's Complete Practical Treatise on the Nature and Use of Logarithms and on Plane Trigonometry (5s., Key 3s., Thomas Laurie, Edinburgh). Demonstrations are "introduced only in so far as they were thought to be necessary towards the understanding, retaining, and judicious handling of the practical processes." The explanations are very simple, the practical nature of the work is well sustained, the treatment of Logarithms and their numerous applications is full of illustration and useful comment, and there are above 80 large octavo pages of logarithmic and trigonometrical tables. Professor Elliott is also the author of an Elementary Treatise on Logarithms and Trigonometry (2s., Thomas Laurie), which less advanced students of practical Mathematics may profitably read.

Kimber's Mathematical Course for the University of London (12s., Key to First B.A. Part, 5s., Longmans). We will take this opportunity of discussing the value of a work concerning the utility and scope of which many inquiries are made. It professes to "contain an outline of the subjects in Pure Mathematics included in the Regulations of the Senate for the Matriculation and First B.A. and First B.Sc. Pass Examinations," and the author states in his Preface that "the object of the present volume is to economise the time and energies of students preparing for the [above] examinations, by presenting in one volume the information they must otherwise collect from various sources. It includes the subjects in Pure Mathematics prescribed for those examinations." We cannot understand this statement when we consider that the Regulations under the head of Geometry prescribe four distinct subjects in Pure Mathematies which are not touched upon at all in the above work. We refer to the subjects referred to and described on pages 197 to 239 of the present work. Under the same head are also mentioned the Elements of Co-ordinate Geometry," obviously including Polar Co-ordinates, which are not explained in Mr. Kimber's book. But with this latter omission we find no fault, seeing that this change in the Regulations had not come into effect when the last edition (the fourth) was issued. Our frank judgment is that though in its several parts the work has many excellences, yet, taken as a whole, it would hardly be safe for the self-taught student to trust himself unreservedly to it. Nevertheless, we may say that for comparison and use with other works; as an indication of some of the essential points in each subject treated upon; as a guide to the general character of the questions set from year to year; for use with a tutor who could give the student the more extended applications

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of which each subject is capable; and as a repertory of most of the questions in Pure Mathematics set by the different examiners from 1838 to recent years, it is a valuable work. But in all cases the last edition should be procured, for the author continues to improve and add to it from year to year. Of late many solutions and additional examples-some of which might be more concisely and accurately worked-have been added, especially in the chapters on Conic Sections. In Arithmetic, the Sections on the Divisibility of Numbers, Prime Numbers, and Terminating Decimals, are very suggestive; and in Plane Trigonometry, the deductions of formulæ, solutions of triangles, and measurements of heights and distances, will serve as a useful guide to the beginner if resorted to in cases of actual difficulty, and not as a temptation to the mental indolence which destroys all self-reliance—that indispensable virtue in the candidate for a University degree.

In Note A of the New Edition there are both algebraical and geometrical solutions of the problems which relate to the formulæ for the ratios of the sum and difference of two angles, whatever be the magnitude of the angles. These solutions will be of use to the student who may find a difficulty in dealing with particular cases. In spite of the drawbacks we have mentioned, many a candidate will fittingly find a reason in what we have said for procuring a copy of Mr. Kimber's helpful work.

GEOMETRY AND TRIGONOMETRY.

FIRST B.A. AND FIRST B.Sc. PASS EXAMINATIONS.

1872.

Examiners, Prof. H. J. S. Smith, M.A., F.R.S., and Prof. Sylvester, LL.D., F.R.S.

ARITHMETIC AND ALGEBRA.

Morning Paper.

1. State and prove the rule for the division of decimals.

If the length of the year is 365 242264 days, but is reckoned as equal to 365 days, find in how many centuries the accumulated error would amount to 263 days.

2. Find the greatest common measures of 11310 and 86478, of 86478 and 448630, and of 11310, 86478, 448630.

Find the least common multiple of 103, 63, 4%.

/147 (√5+√/3 _√/5−√/3
6055-√

3. Simplify

Express the fraction

rational denominator.

4. Find the value of

(a2+b2-c2) (a2+c2−b2)+(b2+c2 − a2) (b2+a2−c2)+(c2+a2−b2) (c2+b2—a2). (a+b+c)(a+c−b)(b+c−a)(a+b−c)

5. If four quantities are proportionals, and the second of them is a mean proportional between the third and fourth, prove that the third will be a mean proportional between the first and second.

6. Find the number of Permutations, and also the number of Combinations, of n things taken m at a time.

Wit 17 consonants and five vowels how many words can be formed having two different vowels in the middle, and one consonant (repeated or different) at each end?

7. Find the sum of a Geometrical Progression to n terms. What is the value of the series -+-....ad infinitum?

8. If an annuity continued for ever is worth 25 years' purchase, what annuity (reckoning at the same rate of interest), to continue for 3 years, can be purchased for £5,000?

A man invests £10,000 in land: he borrows of the value, which he invests as before; he again borrows of the value of his new investment; and so on continually. What would be the aggregate amount borrowed, if this process were continued indefinitely?

9. Solve the Equations

x-y
xy

The sum of three numbers in Arithmetical Progression is 33, and the sum of their squares is 435. Find the common difference.

10. What is the Characteristic of the Logarithm of 50 to the base 2? If the logarithm of 3 to the base 10 is 477121, what is its logarithm to the base 10?

Given log. 648=2.81157501, log. 864-2.93651374, find log. 108.

GEOMETRY, TRIGONOMETRY, CONICS.

Afternoon Paper.

1. If a straight line be drawn parallel to one of the sides of a triangle it shall cut the other sides, or these produced, proportionally; and, conversely, if the sides, or the sides produced, be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining side of the triangle.

2. Prove that if the four sides of any quadrilateral figure are bisected, the four points of bisection are the four vertices of a parellelogram of which the area is one-half of the area of the quadrilateral figure.

3. In a right-angled triangle the rectilineal figure described on the side opposite to the right angle is equal to the similar and similarly described figures on the sides containing the right angle.

4. If two planes which cut one another be each of them perpendicular to a third plane, their common section shall be perpendicular to the same plane. 5. Prove that if two spheres intersect one another, the curve of intersection is a circle of which the plane is at right angles to the line joining the centres of the two spheres.

Given a circle, and a point not in the plane of the circle. Find the centre of the sphere which passes through the given point, and through the circumference of the given circle.

6. OA, OB, OC are three adjacent edges of a cube. Given OA=OB=OC=a' find the solid content of the pyramid OABC, and the area of the triangle ABC'

7. The co-ordinates of the points P and Q being (a, b) and (b, a) respectively, and O being the origin, find the equations of the lines OP, OQ, PQ, and the area of the triangle OPQ.

8. If (x, y) are the co-ordinates of a point P upon an ellipse, of which the x2 y 2 equation is + =1, find at what distances from the origin the axis major a2 62 is cut by the tangent, and by the normal at P; and show that the rectangle contained by these distances is equal to the difference between the squares of the semi-axes of the ellipse.

9. Prove the formula

Sin. (A+B)=sin. A cos. B+ sin. B cos. A, drawing the figure for the case in which A and B are each less than 90°, but A+B greater than 90°.

Find the sine and cosine of 75°.

10. Given in a plane triangle a, B, and A, solve the triangle.

If the sides of a triangle are 9 feet, 7 feet, and 4 feet respectively, what are the sines of the angles of the triangle?