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THE

MONTHLY REVIEW,

For JULY, 1758.

Mathematical Essays; being Effays on Vulgar and Decimal Arithmetic. Containing not only the practical rules, but also the reafons and demonftrations of them; with fo much of the theory, and of univerfal Arithmetic or Algebra, as are neceffury for the better understanding the practice and demonflrations. With a general Preface, including a panegyric, on the usefulness of Mathematical Learning. By Benjamin Donn, of Biddeford, Devon. Teacher of the Mathematics, and Natural Philofophy, on Newtonian principles. 8vo. 6s. Johnfton, Davey, &c.

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ATHEMATICAL Learning, during the laft and prefent centuries, has made a moft furprizing progrefs; and truth, affifted by the uncontroverted principles of that fcience, has banished hypothetical chicanery from the regions of philofophy. It is therefore no wonder, that a great variety of Authors, defirous of extending fo valuable a branch of fcience, fhould have written on every part of Mathematical Learning. But ftill, a general courfe of Mathematics and Natural Philofophy, tracing the fcience from its first principles, and exhibiting the demonftrations on which each rule or problem is founded, is ftill wanting; there being none in cur own language that can, with any fhew of juftice, be called a Course of the Mathematics and Natural Philofophy, according to the modern improvements, and properly adapted to learners.

This defect Mr. Donn has undertaken to fupply; and the work before us is intended as the first volume of a Courfe of Mathematics and Natural Philofophy; it contains the whole VOL. XIX.

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science

fcience of Arithmetic, both with regard to theory and practice, and is introduced with two prefaces; one general, and the other particular. In the general preface the Author endeavours, ' 1. To fhew the dignity of the Mathematical Sciences. 2. Their ufe to men in general, in the improvement of the mind. 3. The advantage of thofe fciences in fome particular profeffions. 4. Laftly, to make fome general inferences by way of conclufion And in the particular preface, Mr. Donn has given. a concife hiftory of Arithmetic, and a catalogue of the principal Writers on that science.

The method made ufe of by Mr. Donn in thefe Effays, is, firft, to lay down the neceflary definitions and axioms; fecondly, to give the rule, and illuftrate it with examples; and, thirdly, to add its demonftration, which is generally in a note at the bottom of the page. The demonftrations are chiefly algebraical, as being the fhorteft, and eafieft to be understood; and, indeed, in fome cafes, the only method that can be taken: many of the operations in Arithmetic being incapable of demonftration by any other manner of reafoning. Thofe, however, if, indeed, there are any fuch, who will reft fatisfied with bare rules, without knowing the foundation on which they are built, may be accommodated with Mr. Donn's treatife; for the demonftrations may be passed over, without any confufion to the Reader, being, as it were, feparate articles from the text. But we hope few will be fo blind to their own intereft as to read thefe effays in fo careless and fuperficial a manner: for no perfon can be faid to understand any rule in Arithmetic, if he is ignorant of the reasons on which it is founded and it is furprizing, that fo great a variety of books fhould have been written on this fcience, many of them by perfons undoubtedly equal to the task, yet all, Mr. Malcom's treatise excepted, deftitute of the demonftrations of the rules given for performing the feveral operations.

It is a general complaint, that few pupils make any tolerable progrefs in Arithmetic during the time of their being at school; and that the generality, after fpending feveral years under the tuition of fome mafter, are almoft totally at a lofs to apply the few rules they have learned to the real ufes of life. But a fmall degree of reflection would fuficiently convince us, that any great progrefs is not to be expected from the common method of teaching. The child learns a few rules by rote, which are therefore foon forgotten; while the moft material particulars, the reasons on which thofe rules are founded, and their extenfive ufe in the various concerns of life, are totally omitted. And surely the learned themfelves cannot wholly be acquitted of having contributed, in fome measure, to propagate fo prepofterous a method

of

of teaching for by publishing rules without the reafons en which they depend, the generality have imbibed a notion, that demonftrations are ufelefs in all cafes, and in fome impoffible to be given. The work before us will therefore be of the greatest ufe in removing thefe errors, and pointing out a method of ftudying Arithmetic in a rational and scientific manner.

At the end of each rule Mr. Donn has given the most useful contractions, together with all the improvements lately made, in order to fhorten the feveral operations. Thus, in Multiplication he has not only given the common contractions, but also shewn the method of multiplying feveral figures by feveral, in one line: an invention which lately made a great noife among Arithmeticians.

After fhewing the nature of the rule of proportion, generally called the Golden Rule, the reasons on which it is founded, and its extenfive use, our Author has added the following obfervations, which deferve to be attentively confidered by every ftudent in Arithmetic; as they tend to prevent his committing errors otherwise almost unavoidable.

The rule of proportion being very extenfive, and there being innumerable ways of propofing a queftion, it may be fo complicated, as many times to require a confiderable judgment to know what things are proportional, in order to ftate the queftion; and for thefe reafons it is impoffible to give any general direction, that fhall reach all cafes; for, after all that is, or can be done, the bringing queftions out of the complicated <language of the queftion, into numeral expreffions, must chiefly depend on the judgment of the Arithmetician; all that can be done to help the young Arithmetician, is to propofe • fuch a variety of questions, as, when he becomes mafter of them, it may be fuppofed he will be able to folve any other that may fall in his way.

• Before we put an end to this chapter, it may be proper to <hint to the young Arithmetician, that it is abfolutely neceffary, before he states the queftion, to confider whether the terms are in direct proportion to each other; for, otherwife he may commit grofs errors, by taking fuch things to be in fimple proportion, which are not fo; thus, though in buying and felling, the price of the goods increases or decreafes, in the fame proportion with the quantity of the goods, yet in geometric, philofophic, &c. cafes, thofe things which at firft fight may, to many perfons, appear to be in fimple proportion to each other, may not be fo upon mature confideration; wherefore, fuch perfons as would folve fuch queftions, muft firft acquaint them⚫ felves with the laws thereof; the neceffity of which knowlege B 2

• may

may be fhewn by an example. Let us fuppofe then, that there are two towers, one is fixteen feet in height, from the top of which a stone being let fall, fell to the ground in one fecond of time; it is required to find how high the other tower is, from which a ftone falls in three feconds?-Here, a Tyro may conclude, that, fince the higher the tower is, the longer time the ftone must be in falling, that the fpace the ftone falls through, will be in fimple proportion to the time; and therefore would ftate the queftion thus, as 1": 3" :: 16 feet: 48 feet, for the height of the tower, which was required; but if he afks a perfon acquainted with the laws of falling bodies, he will be informed, that falling bodies do not fall equal fpa'ces in equal times; but that, the greater space a body has fallen through, the greater is its velocity; and that the question ought to be thus ftated, " X 1: 3′′ X 3 :: 16 feet: the answer, or as, 19: 16 fect:: 144, the true height of the tower 6 as was required.'

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A caution of the fame nature is given in the next chapter, which treats of reciprocal proportion, or the Rule of Three Revere. As a learner,' fays our Author, may be apt to take things in fimple direct proportion which are not fo, as we have already hinted, fo in this rule, if he does not reason with himself, before he states the question, he may take fome things to be in fimple reciprocal proportion, which are not fo; for example, fuppofe that in a room, where two men, A and B, are fitting, there is a fire, from which A is three feet, and B fix feet diftant, and it is required to find, how much hotter it is at A's feat than at B's? In folving this queftion, at first fight, the learner thinking, that as it is evident that the nearer a perfon is to the fire, the greater heat he muft feel, may conclude, that this is a queftion in the Rule of Three Reverse, and therefore to be ftated thus: if 6 feet: 1 degree of heat :: 3 feet reciprocally: 2 degrees of heat; or that the heat is twice. fo great at A's, as it is at B's feat: But let the Tyro go to a Philofopher, a perfon who is acquainted with these things, and he will be told, that, according to the principles of Philofophy, it fhould be flated, if 6 X 6: 1 :: 3 X 3 reciprocally, or as 3X 3: 1 degree :: 6 X 6 directly: 4 degrees of heat, or that it is four times fo hot at A's feat as at B's. Whence it appears, that in folving fome queftions which may feem to belong to common rules of Arithmetic, there is not only required the knowlege of Arithmetic, but also of fome "other science.'

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We fhall conclude our account of this treatife, after recommending it to the perufal of those who are defirous of attaining

a thorough

a thorough knowlege of Arithmetic, with the following demonftration of the common rule for extracting the fquare root.

It is time now to proceed to the demonflration of the rule delivered in article 459, for extracting the fquare root; and, for the more regular doing this, it may be proper to premife,

First, that the number of figures in the product of any two numbers may be equal to, but cannot poffibly be greater than the number of figures in the two factors: and, though they ⚫ may be lefs, yet never less than one fewer.

Demonftration. That the product may have as many places of figures as are in both the factors, one example, for inftance, 9X7=63, is a fufficient demonftration. But that the num⚫ber of places, in the product of any two numbers, cannot be more than the number of places in both factors, we thus fhew: Let a and be the two factors, p their product, c = a greater ⚫ number than b, viz. = with as many o's on the right hand • as there are places in b; then it is plain, that a c must be greater than ; but a ca with as many o's on the right hand as there are places in b; . the number of places in a c is number of places in a + the number of places in b; whence the number of places in a c, which is a greater product than p, is only the number of places in a and b; and, confequently, which is lefs, cannot have more, for that is ab

furd. 2. E. D.

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As to the above affertion, that the number of places in the product may be one lefs than the number of places in both ⚫ factors, one example, viz. 2 × 3 =6, is a fufficient proof.

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It only remains now to be demonftrated, that the number ' of places in the product of any two numbers cannot, in any cafe, be less than the number of places in both factors minus ' one. And this may be thus demonftrated: if one lefs than 'the number of places in each factor is denoted by d and n refpectively, the leaft numbers, confifting of the fame number of places as the factors, are 1 with d, o's; and I with n, o's, C by the nature of notation; and their product is I with d, o's +n, o's, that is, it confifts of 1+d+n places; but d+1, ‹ and n + 1, being the number of places in each respective factor, their fum, or the number of places in both, is d+n +2, which is but one more than the number of places in the ⚫ above product (1+d+n); therefore the leaft factors poffible can have but one place lefs in the product than the number of places in both factors, and contequently no other factors, for the leaft factors must have the leait product. Q.E.D.