tivated by the French and German mathematicians. The change was no doubt for the better as a method of discovery, but the close and grasping character of the Greek Geometry did not accompany that of Descartes. Though not less certain in its results, the algebraical method was rather a mechanical art than a science, and its employment does not require the same expenditure of thought as the purely geometrical. Hence the student ought always to have enjoyed the benefits of a regular course of study in the ancient Geometry before he is introduced to the more mechanical and less practically logical, though most valuable, system treated of in the volume before us. As a quaint editor of Euclid has observed Algebra is the paradise of the mind where it may enjoy the fruits "of all its former labours without the fatigue of thinking." In Britain the case was different; Newton had contracted a great admiration for the methods of the ancients, and, after investigating and discovering his beautiful theorems by the Cartesian analysis, threw the proofs into the guise of the synthetic method. The effect of this was for a long time to check the spread of co-ordinate Geometry in Britian; the British mathematicians seemed to consider it heresy to depart from the method which Newton had sanctioned; and even in the present century Sir John Leslie attempted in his Geometry of Curve Lines' to supply by the old methods the demonstrations of propositions, for which such methods had long been felt to be inefficient. During the last forty years, however, many systematic works on this important department have been published; and every few years is adding considerably to their number. This is of itself a proof that there is a demand for such works; and in this highly practical age it shews that the spread of a knowledge of this branch is necessary for the preparation of engineers and other professional men for the business of life. It has also found its true place in a course of sound and useful education, not as a substitute in any form for the more logical and rigorous method of deduction, but as supplementary to it, and to which the older is absolutely necessary as a preliminary. It seems not a little surprising that the learned gentlemen who drew up the regulations for the Calcutta University, should have entirely passed over a branch of pure science of such universal application in every department of physical research-without a competent knowledge of which no student can read the ordinary standard works on mathematical mechanics, astronomy, &c. No doubt it is included in the extra subjects proposed for M. A., but the host of subjects of the highest difficulty mentioned for honors in mathematics is such as may well discourage any student from attempting to master them rather than those proposed in other departments.

Mr. Smith's Geometry, the author tells us, was first undertaken during a period of protracted indisposition, and a voyage to and from the Cape of Good Hope, in 1841-2. His object was that of providing for his classes in the Free Church of Scotland's Institution in Calcutta, a text book in Analytical Geometry better suited to his views than any he then found procurable. From time to time subsequently he added to and corrected the manuscript, and finally published it early last year when on a visit to Britain. This work then, though

recently published, seems to have been written previously to some of the treatise, now most generally used in Britain. The author first discusses analytically the properties of the straight line, circle, ellipse, parabola and hyperbola, and then in an additional chapter treats briefly of the application of the analysis to some of the curves of higher degrees than the second. The treatment of the whole is simple involving only the ordinary operations of Algebra. This renders the work one of ease to the beginner, and enables him to commence at a much earlier stage than such works as require some knowledge of the differential calculus. This however is not a peculiarity of Mr. Smith's work, for though Dr. Lardner makes use of the calculus, Mr. Todhunter and others have discarded it from their elementary works. Much of Mr. Smith's method seems to be his own; the equations he requires, he almost always derives in the first case in the most general form, and thence shews how the particular cases flow from it; this, though not the most encouraging at first sight, has its own advantages, and is calculated to give the learner an ultimate command over the subject not attainable by every method. In the course of perusal there are some things that appear as defects or omissions, as well as things in his methods of solution which, if not always very elegant, are at least somewhat original. The decided want of the book is the entire omission of polar equations, which however different from the method of co-ordinates, is both necessary to a thorough knowledge of analytical plane Geometry as an instrument of investigation, and to enable the student to read with intelligence and appreciation works on physical science, in which operations of an analytical kind are frequently employed. Though the author has made no reference to such as Pascal's theorem and Transversals, these may be considered as beyond the aim of the treatise. In a work of the kind we should have expected to find definitions of such terms as "equation of the first degree" and "tangent," before using them. The position of the single unknown quantity on the right side of an equation, instead of the left, is not elegant, and the expression concourse of a line with a circle" (p. 28) is scarcely in good taste. Such slips as these, and the omission of the necessary limitation of the conditions of the problem in § 126, are only such as may be expected to occur in first editions, as must also errata in the printing of what was correctly written.*

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It might have improved the work, had a collection of exercises been appended to each chapter; for, of whatever value the explanations of a well written text book may be to the learner, the lessons are best imprinted in his mind by being put in practice. On the other hand there are many problems discussed of such a nature as are likely to awaken young minds of a mathematical cast, and set them a thinking for themselves; such are those in §§ 136 and 210,

* From these Mr. Smith's treatise is not free, though they are generally such as may be easily corrected; thus, in the notes (p. 31) we find "an elliptic alone" for "an elliptical one;" on page 12, (line 3 and last line)", ', for "—'; and in $239, atangent to the ellipse," when "a tangent to the hyperbola." was evidently meant.

which by the bye might as well have been grouped together, and those in §§ 387, 388,* 390 and 409, the last being Newton's method of trisecting an angle by the conchoid. The want of abridged forms of notation can scarcely be considered a defect in a work intended only or chiefly for elementary instruction. The abridged notation is often a stumbling-block to beginners, and when a good elementary work like the present has been mastered, it is easily acquired.

We only hope the author may meet with such encouragement in the sale of the work, as may reward him for the pains he has bestowed upon it in order to provide a good introduction to Analytical Geometry of two dimensions.

Narrative of the Campaign of the Delhi Army. By Major H. W. Norman, Deputy Adjutant General of the Bengal Army. London; W. H. Dalton, Cockspur Street. 1858.

THIS is an official report, published by permission of the Governor General. Its author went down in May with General Anson from Simla on hearing of mutinies at Meerut and Delhi. He was then Assistant Adjutant General. In a few days the Commander-in-Chief died, and a new man took command who was totally unacquainted with Indian warfare. In the first engagement Colonel Chester, the Adjutant General, (one of the few men who cheered on others in that great crisis,) perished in the field. Henry Norman then acted as Adjutant General. He was relieved by Brigadier Chamberlain who very speedily was laid aside by a painful wound, and was unable to resume charge of his office till the day of the assault. During all the interval, amidst the changes of commanders, with sickness, and the fire of the enemy, slaying hundreds, Henry Norman performed the duties, and exhibited throughout such Christian courage, and such consummate judgment and temper, that at the end of the operations he was recognized by the whole force as the foremost staff officer of the army. As soon as Delhi was taken he marched down with Brigadier Greathed's column; then joining the Commander-in-Chief, rode by his side throughout the relief of Lucknow. He was made Deputy Adjutant General, with the official rank of Major, and continued at his post till compelled by a wound, received at Bareilly, to seek rest for a month at Umballah. Like Sir Hope Grant, and a few more, he had been engaged almost without intermission for thirteen months, and in that time had been more than fifty times under heavy fire.

His Narrative never alludes to his services, but there is not a Delhi officer who disputes his claim to eminent distinction. Many speak

The problem in this section may serve as a specimen :-"To find the figure which the extremity of the shadow of a vertical pole traces out during the day, on a horizontal plane." If with Delambre, we consider that as the sun in the course of each day describes a parallel of declination, a ray passing just over the top of the perpendicular pole will describe a conical surface, it is easily seen that the intersection of this surface with the horizon must be a conic section. Mr. Smith's analysis determines under what conditions it is any particular section.

with enthusisam of his uniform cheerfulness, his constant zeal, his conspicuous example; but he never speaks of himself in complaint or in boasting. He is a subaltern still, because his regiment, (the 31st,) was loyal, and did not kill its senior officers; and the Queen according to etiquette "cannot" make him a brevet Major, a C. B., nor "can" she thus honor Willoughby, Osborne of Rewah, nor Charles Nicolson who led the gallant Probyn into action; nor "could" she thus reward Salkeld or Willoughby if they were living; and thus etiquette dooms us, as far as in it lies, to worn out Brigadiers and helpless Colonels. A man like Charles Reid of the Sirmoor Battalion would have been in command of a division under Napoleon. Services like his, with his invincible hand in his invulnerable post, (never once, in the long four months of seige, leaving his post to visit the camp,) would have caused him to be prized as a hero and rewarded worthily. But in the English service he finds himself now disabled by his wound, and as a Major and nothing more, must probably relinquish active service. No one grudges Probyn, young as he is, his Brevet Majority and Companionship of the Bath. But he owes them not to his merit only, but to his merit combined with the fact that he became a Captain through casualties during the mutiny. His Commander in the second Punjab Cavalry (H. John Nicolson's brother,) a man of longer standing in the service, is still Lieutenant Nicolson, and bears about him the permanent mark of his sufferings and his devotion, and the consciousness, that, thanks to etiquette, he has been superseded. Cases of this kind might be multiplied. Perhaps in Major Norman's this hardship is less felt, because his official position opens prospects to him which etiquette itself cannot destroy.

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The pamphlet before us was published at home when the public was absorbed by Lucknow, yet certainly no one who understands recent events will deny that the Delhi army both did most, and endured most, in the campaign of last year. Sir Archdale Wilson's despatch was a worthy record of its exploits. We have since seen many elaborate productions which have narrated far less important operations far more ornately. Indeed it seems to be the fashion to despatch" despatches only after four or five months have been devoted to their composition. The taking of Jhansi was very ably described, but the tale we apprehend might have been narrated the next week. And if the capture of Culpee is strikingly described, we may be permitted to say that six months were not required to prepare the picture. And so with respect to results. General Franks killed, he informs us, 800 men without losing one of his own. Other officers who were Thus our conpresent reduce the number of slain very considerably. fidence is somewhat shaken; and we turn with no slight comfort to Sir Colin's despatch on the Lucknow Relief, and to this Duke of Wellington sort of plain unadorned narrative of facts. We find notices of simple incidents scattered about, without a single word of comment, and we feel that they need none. Thus, when the siege began, the whole of our force consisted of 600 Cavalry, 2,400 Infantry, and 22 The cholera was never field guns, with a weak siege train.

absent from the camp; on the day of the assault there were

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in camp no less than 3,074 sick and wounded. The enemy were represented to the English public by such well informed statesmen as Mr. Vernon Smith, as shut up in "Delhi Fort," and destined to be overwhelmed in a few days. Major Norman quietly says "Suffice it to say that a wall twelve feet thick, with a 'ditch in front of considerable width and twenty-four feet deep, with an admirable glacis covering the wall for a full third of its height, bastions in capital order, each holding ten, twelve or fourteen pieces of heavy artillery, so as to form flanking defences around a city seven miles in extent, with the river on one face, constitute a formidable position," "On the morning of the 9th June the Guide Corps consisting of there troops of cavalry and six companies of infantry (rifles) marched into camp under the command of Capt. Daly. This distinguished body of men had marched at the hottest season of the year from Murdan on the Peshawar Frontier to Delhi, a distance of 580 miles, in twenty-two days, and though the infantry portion were 'occasionally assisted with camels or ponies on the line of road, the marching was a surprising heat, even for cavalry," "The truunions of one eight-inch howitzters giving way one after ancther, these pieces were sunk in the ground and used as mortars, their shell ' having a very long range. In our batteries we used the enemies' ' ordnance captured at Badlee, and as we had no ammunition for the captured twenty-four pounders, though that of this calibre fired by the 'enemies were picked up and sent back again." "At the lowest estimate the numbers of the insurgents were 30,000 men. These guns, as we know, were numerous as even they could have desired, and 'their ammunition appeared inexhaustible. The force was insufficient to invest event one third of the place, and access to the left bank of the Jumna was at all times perfectly secure, by the bridge'of-boats, which was under close fire of their ordnance at Selinเ gur, and fully 2,500 yards from our nearest guns. We were there'fore powerless to prevent a constant stream of re-inforcements, and supplies from pouring into the city, and were thankful that we had been so far enabled to keep open our rear and freely to communicate with the Punjab, whence all our resources well drawn. Had the numerous cavalry of the insurgents been directed with judgment and 'boldness, it is not too much to say that we should have been put to the most serious straits." "The rifles commenced with 440 of all ranks; a few days before the storm they received a re-inforcement ' of nearly 200 men; their total casualties were 389. The Sirmoor 'battalion commenced 450 strong and was joined by a draft of ninety · men. Its total casualties amounted to 319. The guides commenced 'with about 550 (cavalry and infantry), and the casualties were 3033."

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This kind of military writing is worth volumes of poetry about "their red lines" which in fact were never approached by the Russian Cavalry, and the Highlanders assulting first breaches that were really carried by others, and any numbers of campaigning letters full of word-painting, in which men draw on their excited imagination for their facts. And in like manner this siege of Delhi will prove to have been a feat of arms with which none