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Mathematical Tables, &c. By Peter Barlow, of the Royal Military Academy. London: G. and S. Robinson, 1814. IN calling attention to a work which is out of print, we do not think we transgress the legitimate bounds of our undertaking. However desirable it may be that new productions of merit should be brought under the eye of the reader, he should not be allowed to forget, that many books of unquestionable utility, which, from the nature of their subjects, are only in the hands of few, glide gradually out of memory, until at last it is seldom that a copy can be obtained. The tables now before us are in this predicament. They nevertheless well deserve to be reprinted, as an examination of their contents will show.

On looking for any number under 10,000, we find in one row, its factors, square, cube, square-root, and cube-root to seven places of decimals, and its reciprocal to ten places. This is in fact the body of the work, and is a most superior table, both as to the contents, and the accuracy and arrangement with which it is printed. It supplies, in some measure, the place of a common table of logarithms; and is, for many purposes, of greater utility. It is followed by a table of the first ten powers of all numbers not exceeding 100, and of the fourth and fifth powers of all numbers from 100 to 1000. This is followed by a table for facilitating the solution of the irreducible case in cubic equations, and by a list of all prime numbers from 1 to 100,000. We have then Naperian logarithms from 1 to 10,000 to eight decimal places, followed by the value of the first six co-efficients of the nth power of a binomial, for every value of n from .01 to 1.00. The work ends with a collection of formulæ in different branches of mathematics; and a comparison of the weights and measures of different nations.

That such an undertaking, if well done, must be of the highest value, need be told to no one, who has experienced what we beg leave to call the nuisance of long arithmetical calculations. That the work before us is well done, we know from experience. It would hardly consist with our purpose to enter into the details of the methods by which its different tables were constructed, since it matters little to the reader. We hope, by calling attention to the merits of the book, to promote its being reprinted; and with this view we will show how much trouble it might save to schoolmasters who have it in their possession, believing it superfluous to insist upon its utility to the calculator.

It is well known how necessary it is that many examples

should be worked by beginners in arithmetic. The common books on this science usually contain a moderate number; but in addition to the frequent errors of print which are found in the answers, the idle may, and if the customs of our time still prevail, do, obtain assistance from the more industrious, which it is difficult to prevent while all are, or have been, employed upon the same questions. In Mr. Barlow's tables, we have the means of avoiding this difficulty in many rules; since, on a rough computation, we have 20,000 examples of common multiplication, as many of division, extraction of the square and cube roots, and innumerable examples of the rule of finding the greatest common measure, and least common multiple; and all these directly given with their answers; while the labour of half an hour would furnish an expert arithmetician with the means of setting a whole school to work for a day on almost any subject. Examples of addition might be obtained by recollecting, that the sum of the cubes of all numbers from m+1 to n inclusive, is (≥ n. n+1)2 -(m. m + 1), which may be readily found from the tables. The mathematician will be able still further to avail himself of the various relations which exist, for the purpose of multiplying instances; and we may safely say, that he will thus have at his command a greater number than have been contained in all the works of arithmetic that ever were printed. The tables of weights and measures, at the end, would very much expedite the formation of examples in the commercial rules. The principal instrument, however, the table of squares and cubes, does exist in several different works. For example, in Hutton's Tables, printed by the Board of Longitude, reviewed in the first number of this Journal, and by themselves in a little work printed anonymously at Paris, entitled Tables des Nombres quarrès et cubiques, et des racines de ces nombres, depuis 1 jusqu'a 10,000.' For the accuracy of this we cannot vouch.

There is one observation to be made upon the method of printing Mr. Barlow's tables, which applies to most similar works lately executed in England. Calculators themselves have insisted but seldom upon any particular form of the figures; and the printers, who look at the beauty of the whole, and not at the manner in which the several parts can be distinguished from one another, have introduced the practice of cutting figures all of the same length, which is more consonant, we suppose, to their notions of typographical elegance. Thus 6, 7, and 9, have taken the place of the old six, seven, and nine. In our opinion, and we believe we are joined by most of those who have much occasion to use

tables, this is far from being an improvement. The figures become undistinguishable: thus in using a table rapidly, the 6 and 9 are apt to be confounded with 0, and the 3 with the 8. It is to be wished that the old forms were restored, which would be done speedily, if the example were set in widely circulated works. Thus the Nautical Almanac, the Requisite Tables which are now preparing, the transactions of Societies, particularly the Astronomical, might be made instrumental in effecting the change, if their directors were of opinion that it would be advantageous.

This improvement might we think be adopted without increasing the breadth of the type, so that the same number of figures would be contained in a line. Of this, however, we are not certain, not knowing whether the distinctness of the ancient figure may not, in some degree, be owing to the greater room allowed to it. Thus in Vlacq's Logarithms, 1628, and Dodson's Canon, 1742, there are only twelve figures to the inch; while in Barlow's Tables, 1814, there are nineteen, and in Babbage's Logarithms, 1827, there are sixteen. The subject deserves particular consideration, since all circumstances which may cause a wrong figure on the calculator's paper are of equal importance; and if it be worth while for a man of talent to form tables, and to employ a good printer in composing them, and an industrious corrector to prevent mistakes, it is just as well worth while to use a type which shall prevent their joint labour from being misunderstood by the man who is to profit by it.


The Etymological Spelling-Book and Expositor, being an Introduction to the Spelling, Pronunciation, and Derivation of the English Language, &c. &c. By Henry Butter. Fourth Edition, London.

WE should hardly think it worth while to discuss the merits of any work known by the name of a spelling-book, if it did not differ materially from those in common use. Perhaps many of our readers may still recollect those days of early misery in which they were compelled to spell, as it is called, words known by the awful name of polysyllables, without comprehending in the slightest degree the meaning of these mysterious sounds. Spelling is still an important part of early education in elementary schools. It enters into the items enumerated in the parallelogrammical* boards that are

*This is one of Mr. Butter's difficult polysyllables. See his book, p. 24, and the present article, p. 162.

sometimes placed as an advertisement in the windows of the humblest class among the instructors of youth; and even in the houses of the better educated we may occasionally see a careful mother hearing her children spell, with a gravity suited to the supposed importance of the subject.


Let us consider what is the object proposed to be attained by spelling lessons. When a child has learned to read (which, according to the usual system of teaching, is the most difficult attainment of his life), he has acquired the power of expressing the sounds, which the printed words placed before the eye are intended to represent. The child, in fact, gives a name to each printed word that it sees, just as it gives a name to the picture of any known object. When a child has seen a real horse, and has heard the name horse' given to the animal, the same child will readily apply the same name of horse' to a correct picture of the animal. This is the first step towards the understanding of signs that represent things. It is the first step, or ought to be made so, towards learning to read. First, the real object is made known, and its existence and most striking properties are associated with a certain sound; as in the example just taken, the sound 'horse' and the animal horse' become connected in a child's mind as one thing. Secondly, the sound is transferred also to a picture of the object which resembles the thing. Thirdly, in learning to read, the sound is connected with certain signs called letters, which have no resemblance either to the sound or the thing signified. The sound horse' does not resemble either the word 'horse,' or the picture of a 'horse.' But by frequent practice we learn that a certain set of letters is to be connected with a certain sound, and no other sound, and this sound reminds us of some object which we have seen either in reality or in a picture*. It will easily be seen from these remarks, that we propose to teach children to read by beginning with short words that have a meaning, and not by beginning with the letters of the alphabet.

One object of spelling is, that a child may know what signs or letters enter into the composition of each word; in fact, that he may know how to write a word. And to know how to write words, what exercise is so appropriate as the practice of writing words? One great cause of the continuance of spelling-lessons is this. The child is first taught to give names to the letters, as a, bee, see, double u, &c.; but when he comes to spelling and reading (for

*This of course only applies to those words that represent visible objects. Words that denote other qualities and abstractions belong to a more advanced stage of instruction.

OCT.-JAN. 1832.


reading is commonly taught by spelling), it is found that the names of the letters are not the same as their sounds when they form words. Hence to learn the word wine, a child is taught to say double u, aee, en, ee, wine; but he is not taught that double u, aee, &c., are merely names for the signs composing the word wine, and not the actual sounds of those signs as they exist in that word. Hence arises a perpetual confusion between the names of letters and their sounds; and hence the necessity, as is said, of spelling-lessons, that the pupil may know how to write those words, whose pronunciation differs from the spelling; in which catalogue, according to the present system of spelling, we must include every word in the language.



We, therefore, object to the whole first part of Mr. Butter's spelling-book, if used, as he recommends it to be, in the way of being repeated several times.' This first part begins with easy words of three syllables, accented on the first syllable,' arranged alphabetically. We see no advantage whatever in the alphabetical arrangement of such words, and we see great disadvantages. If they were arranged according to the final syllables, instead of the first letter, then such words as implement' rudiment' and 'settlement,' would all come together, and each word would be just as readily found in its appropriate column by referring to the last letter as to the first. Mr. Butter, after giving easy words of three syllables, and polysyllables, arranged according to their accented syllables, proceeds to difficult monosyllables,'' dissyllables,' &c., arranged similarly to the first class. There is one advantage in the mode in which he has arranged these words; they are classed according to the accented vowels (in addition to the alphabetical arrangement), and stand thus, beginning with the monosyllables :


Like a in Fate.

in Far.

in Fall.
















The rear of this list of difficult words is of course brought up by those stout soldiers, the polysyllables. Here are a few of them; we should be afraid to present more than six or eight at a time :—

ipecacuanha parallelogrammical septentrionality

As far as we understand Mr. Butter's plan, all the words of this first part, both hard and easy, are to be learned or




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