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lution of numerical equations of the third and fourth degrees very naturally follows the discussion of the particular examples alluded to, and here we have a sketch of Horner's beautiful process of development of the roots of equations exhibited in a very simple and intelligible form. It is well known that the discovery of the initial figure of each of the roots of a numerical equation is the only thing of any difficulty connected with the process of solution, and the author has supplied a new method for the discovery of the leading figure of a root, which affords material assistance to the learner, and “which will always prove acceptable whatever other aid may be combined with it."
The common process of division is merely the solution of a simple equation, as ax = b, while that of the extraction of the square root is the solution of a quadratic, as ra, and hence we see that these processes are intimately connected with the general solution of numerical equations, and are only particular cases of that elegant and general process discovered by Horner about thirty years ago, a process of so much utility that every student ought to be thoroughly acquainted with it. The theory of proportion and progression, and the method of solution of simultaneous quadratic equations are given. in chapter x., and the concluding chapter is devoted to the binomial and exponential theorems, and the theory of logarithms. A proof of the binomial theorem is here given, but it is restricted to the particular case of the index being a positive integer, and for the proof of it in its most general interpretation, the student is referred to the author's "Treatise on Algebra."
On the whole, we regard this neat volume as an excellent introductory work; it is very carefully got up, abounding in original and pertinent remarks. It will be read with interest, and we feel much pleasure in characterizing it as one of the best introductions to algebra we now possess.
The answers to the examples for exercise are kept apart from the work, and given in a separate form, a plan which, in the case of junior students, possesses some advantages.
The first part of Colenso's "Plane Trigonometry" was favourably noticed in the April number of the Journal. The second part now before us is supplementary to the former, and contains the investigations of the various formulæ in the higher parts of trigonometry, together with a large collection of excellent examples in all parts of the subject. The first and second chapters are devoted to the proofs and investigations of several well-known formulæ and series, the special object of the former being "to show the different steps by which we arrive at the value of the number 7, which is of such great importance in trigonometry, and indeed in every branch of mathematics.' oivre's Theorem is investigated and applied to the expansion of sin no and cos ne, in terms of 0, the series of Euler, Gregory, and Machin are then obtained, and from Machin's series the value of π is
determined to a few places of decimals. Machin's series deduced from
is well fitted for the
computation of the value of T, and by means of this and similar formulæ and series, English as well as foreign mathematicians have extended the value of to hundreds of places of decimals. MM. Dase and Clausen have calculated the value of π to a great extent, the former to 210 decimals, and the latter to 250. But our countryman, Mr. William Shanks, of Houghton-le-Spring, has pushed his computations to a still greater extent, having determined the value of π to no fewer than 315 places of decimals. His computations were made by = and
means of Machin's formula, T=40
e', where tan o
π = 4 tan
tan 6' = ; and the value of the number is now being extended, by two independent computers, to 350 decimals.
After noticing Vieta's Property of Chords, and some useful trigonometrical series, the author proceeds in the third chapter to the solution of equations by trigonometry, and to the resolution of x" -1, " + 1, x2-2x" cos 0+ 1, and e* + e* into simple and quadratic factors, deducing also Cotes's Properties of the Circle. In chapter iv. the author finds the sum of certain trigonometrical series, and the following rule of symbolical algebra, namely, "if +r represent a line drawn in any given direction, then a line of the same length, inclined to the former at an angle, may be represented by (cos + √ — Isin 0)r,” is illustrated by some examples of its application.
The two parts together form an excellent work on plane trigonometry, and the large collection of examples appended to the second part will afford ample exercise for the student in every department of trigonometrical research.
GROOMBRIDGE'S FARM AND GARDEN ESSAYS. NO. I. THE CULTIVATION OF ARABLE LAND. Price 4d. (London: Groombridge and Sons, 5, Paternoster Row.)
If the first number of the forthcoming series of essays on the farm and garden may be considered a fair specimen of the character of the whole, we can only say that it will merit a favourable reception. We have carefully perused the publication before us, and hesitate not to pronounce it both good and cheap, and, what is still more important, thoroughly practical. There is scarcely any subject connected with the culture of arable land which is not fully discussed in these pages, and in language simple and intelligible. Whoever desires to become acquainted with the minute details of this branch of farming, of the various natures and qualities of soils, and the different kinds of treatment they require, of the proper times and various methods of fallow
ing, ploughing, and harrowing, &c., will find full and explicit explanations in this essay.
A SELECTION OF ENGLISH SYNONYMS.
(London: J. W. Parker.) We have seen this work, in the pages of a contemporary, boldly called "Whately's Synonyms," on what authority we do not know. Whately appears as editor, and, in the course of the work, the Archbishop's writings are far more frequently appealed to than those of any other writer. It is unquestionably the work of a scholar, confident and discriminating, desiring to take our language simply as it stands, and shaking off its past history as much as possible.
"We have seldom in the following pages," says the author, "introduced what are usually considered so closely connected with the subject of synonyms, as to demand a prominent place in a work of this kind, namely, Etymologies." Again, "Our question is not what ought to be, or formerly was the meaning of a word, but what it now is; nor can we be completely guided by quotations from Shakspeare, or Milton, or even from Addison, or Johnson." . . . "The standard we shall refer to in the present work, is the sense in which a word is used by the purest writers, and most correct speakers of our days." (Author's Preface.)
With such views of the extremely rapid flux of language, perhaps the most useful thing the author could have done would have been to have given us a distinct account of the English words which have changed their meaning since the days of Johnson, or which are now in transitu with what he calls good authorities, to justify his suppositions; this would have been a good excuse for a short work: for, as books of reference, both this and Taylor's of Norwich, on the same subject, recently noticed by us, are far from complete. With regard to these "best authors of the present day," we are reminded of the covenant and
"The purest code
Of churches best reformed abroad;
Though no two of us can agree,
Where or which churches these may be."
Undoubtedly there would be considerable difference of opinion, even amongst men of letters, as to what recent authors may or may not be considered as exhibiting, in their writings, model English; as to whether any deference should be shown to those popular writers, "whose genius can embellish impropriety, and whose authority can make error venerable;' " for such writers are numerous, and influential-Keats, Carlyle, Tennyson, Barrett: whether the chaste "Quarterly," canonizes in its pages, or the more ornate " Edinburgh." Again, with reference to date: " deme unum, atque etiam unum;" if Johnson is obsolete, is his friend Burke, is Paley, is Cowper?
However, the author of the Synonyms does not trouble himself much with quoting modern writers; four or five are all that are ap
pealed to throughout the volume; we are obliged to trust almost entirely to our own recollection, to confirm or contradict the author's dogmas. Were he an ignorant man this would be intolerable, but fortunately he is both well informed and acute, there are no such inaccuracies as occur from time to time in "Taylor's" book; the perception of shades of difference in the meaning of words is amazingly quick and subtle.
"TO TEACH, INSTRUCT, INFORM, EDUCATE" (page 27).
"Teaching,"" strictly speaking, when distinguished from instruction, is applied to the practice of an art or branch of knowledge; instruction to the theory. A child is (correctly speaking) instructed in the grammar of a language, and taught to speak the language, 'teaching' may be merely mechanical; while instruction' implies a degree of understanding in the pupil, as well as the master. A child who has been taught to learn lessons by rote without understanding them, will find difficulty in comprehending instruction in the principles of what he has learned. Hence we speak of teaching a brute, but never of instructing it. In short, a person who is informed, knows something he did not before, one who is instructed understands something he did not before. One who is taught can do something he could not do before."
"DIFFERENT, UNLIKE" (page 51).
"In short, things are said to be unlike,' when they might be expected to be like; 'different,' when non-resemblance is in the natural course of things."
"EFFECTS, CONSEQUENCES, Results."
"Effects,' as distinguished from the other terms we have mentioned, are applied to something which immediately follows from any cause, whether mental or physical. They can therefore, to a certain extent, be calculated on beforehand. Consequences,' are more remote, and spring less directly from causes; they rather follow in the train of an event. We may foresee the consequences of anything, but we always act with a view to its effects. For instance, the effect of wearing clothes, is to cover one; the consequence is that they wear out; we foresee this consequence in buying them, but it is with a view to the immediate effect-the covering us, that we act. 'Results,' are still more remote than 'consequences.' Consequences' and 'effects' are both applied to a change which is in the act of taking place, while 'result' implies the state of things when the change has taken place. Hence it is both more remote and more general."
These are fair specimens, and they certainly remind us of the clear, strong, homely sense and style of Whately, and of his logical keenness. How good again is the following distinction :
"Both these terms are used to express the effect of ill-directed or excessive feeling and imagination; but in romance-the imagination, in sentiment-the feelings have the predominance."
Many of the brief occasional remarks are full of sense, point, and knowledge of human nature. Thus, in connection with the above :—
"Sentimentality is the characteristic of a weaker mind, and is therefore less curable. It is easier to correct an abuse of imagination than an abuse of feeling."
When distinguishing between "meek" and "soft," he observes, "Many think that a woman whose manners are very soft must, necessarily, be meek; whereas, softness is consistent even with self-will and obstinacy." This strongly reminds us of Crabbe's admirable lines:
In some few cases, the author is minute almost to trifling-seeming to hesitate, himself, about the turn of the scale.
"Your friend has found it not an easy thing,
Can from their bent divert this weak but stubborn kind."
“To ‘alleviate' is only used to describe what is done to others; toʻmitigate' is rather oftener applied to ourselves."-Page 29.
We might quote several more instances of the same kind, where good writers would, we are sure, use either of the discriminated words indifferently. Sometimes we think the author is wrong.
"To announce is applied to persons and tidings, but not to opinions; to declare and proclaim to tidings and opinions, but not to persons."
Surely we say with propriety, "Victoria was proclaimed Queen."
Elegant may be applied to mental qualifications, which graceful never
Are we not then permitted to speak of a "graceful style,' a grace
Joyful and its conjugate word joy, are used for the highest degree of pleasure, and always for pleasure excited by some external event."
We do not believe that the sense of the word joy has altered since the days of Pope, who writes thus:
"What nothing earthly gives or can destroy,
The soul's calm sunshine and the heartfelt joy."
"Glad is the lowest degree of pleasure."
He should rather have said, it indicates a lively and self-displaying, not an intense pleasure. "Almost too happy to be glad," is the forcible and beautiful expression of one of our modern poets.
Other instances of the same kind might be pointed out, but we do not wish to pick holes in a work of merit and usefulness.
It is quite in vain for a scholar to attempt to dismiss all the influences of his scholarship. His etymological knowledge evidently haunts the author all through his book, notwithstanding his attempts to make it plain, matter-of-fact, and unpedantic.