

11 ATHEMATICa ARITHMETICS 



[l.NTKODl-lTIOX. 



i, and that it be li- 



ufficient fur the purpose* of 



ope, too, by avoiding all at- 



reerire a full and fail 

 cu*cd to an extent ar 



general education, v f ~ 



tempts at magisterial dignity of style, and addressing 

 our reader* in tbo familiar language of aooial intercourse, 

 to secure thuir attention, and win their confidence. 

 That we may be fortunate enough, by clearness and 

 .. to awaken in omo a genuine 

 love for science, and a desire to prosecute their resen: 

 in writing "f wider scope and higher pretensions, is our 

 earnest desire. 



We hare thought it advisable to commence our work 

 on elementary Mathematics with a preliminary treatise 

 on Arithmetic the groundwork upon which the entire 

 yitem, with the exception of pure geometry, ultimately 

 rests. Books on arithmetic, however, are so numerous, 

 and so easily accessible, that wo might have been held 

 excused from introducing so hackneyed a subject into 

 a work which, though confessedly of an elementary 

 character, is, nevertheless, intended to embrace a range 

 of topics beyond the ordinary limits of a schoolboy 

 course. But" it unfortunately so happens, that books on 

 arithmetic, with few exceptions, are little more than 

 depositories of practical rules and mechanical 

 operations ; and are, therefore, but ill-suited to prepare 

 the young for that higher kind of exertion higher, be- 

 oaose more intellectual which science, properly so 

 called, always demands. 



A boy who has gone through his " Walkingame," and 

 who, generally speaking, is then introduced to "Euclid," 

 is naturally enough bewildered by the total dissimilarity 

 of the two authors not from difference of subject, but 

 of manner of exposition. The former has abundantly 

 supplied him with rules, but no reasons ; the latter gives 

 him rnuoiu, but no rules : the one has loaded his 

 memory, and employed his fingers; while the other appeals 

 to his judgment, and exercises his understanding. 



To make, in this way, the passage from arithmetic to 

 geometry an abrupt transition from the mechanical to 

 the intellectual, we conceive to be a serious defect in 

 educational training. Arithmetic is as much a science 

 as geometry : there is not a rule in the one, any more 

 than there is a theorem in the other, that is not founded 

 on reason, and demonstrably true. And even viewing 

 arithmetic merely in reference to its practical utility in 

 commercial affairs, to the demands upon it in the count- 

 ing-house and the shop wo still contend that its princi- 

 ples should bo rationally taught not authoritatively 

 declared, inasmuch as that which has engaged the un- 

 derstanding, and been received from a conviction of its 

 truth, is more securely retained in the memory than 

 what is committed to it by rote. Rules, unsupported 

 by reasons, are hard to learn, and hard to remember ; 

 but, when the practical precept is associated in the mind 

 with the theoretical principle on which it depends, we 

 learn and remember with ease and satisfaction ; for, 

 instead of words, we get knowledge. 



These considerations. have prevailed with us ; and have 

 determined us to render this course of elementary mathe- 

 matics ooaudtte, by commencing at the very foundation. 

 It must be borne in min>l, however, that our design is 

 not to exhaust hero any subject of which we treat : it is 

 rather to excite an appetite for knowledge, than to 

 satiate it It will not be expected, therefore, that our 

 treatise on arithmetic is to be co-extensive with what, 

 iimler that name, is usually put into a schoolboy's hands ; 

 the bulk of such books arises, in a great measure, from 

 the system of instruction condemned in the preceding 

 observations ; the object of which system seem* to be to 

 inculcate a knack of readily applying rule*, by exp-n- 

 mcnting upon numerous examples, under the guidance 

 of the prescribed directions. This, as the reader has 

 already been made aware, is not our object ; we propose 

 to explain principlet, and to furnish tbo reasons that 

 jiiMify the rules; persuaded, that if the former bo 

 thoroughly apprehended, them need be but little anxiety 

 felt about the mere verbal memory of the latter. 



It wuuld extend these introductory remarks far 

 beyond the space that can be allotted to them, to enter 



into any detailed account hereof the several particulars 

 to be inn.- i.-. >i in this division of the work; hut 

 we cannot conclude tlumi without a few wonla more 

 especially addressed to those who hare resolved to place 

 themselves under our instructions. 



It is a common thing with students in science to be 

 frequently making inquiries as to the . of wli.it they 

 are learning. " What is the UK of this 1" is a question 

 put at every turn ; generally to the annoyance of the 

 teacher, and often to the discredit of the learner. 



The use of any intellectual pursuit employing the 

 term itse in its higher and more honourable significa- 

 tion is to be realised in the mental satisfaction and the 

 mental elevation it communicates. You do wrong to 

 estimate science solely and exclusively in proportion as 

 it visibly contributes to our animal wants and enjoy- 

 ments : there is an intellectual pleasure in the very 

 process of acquiring knowledge ; while the conscious pos- 

 session of it raises the human being in the scale of 

 creation, and thus enables him to contemplate its 

 wonders from a more exalted position. It is in this 

 way that knowledge, like virtue, to which indeed it is 

 allied, is said to be its own reward ; for the study of 

 science is accompanied with gratifications of the purest 

 and loftiest kind ; and is productive of advantages to 

 the student altogether distinct from the benefits con- 

 ferred by its applications to the practical purposes of 

 life. It invigorates and enlarges the faculties retines 

 and elevates the desires and adorns and dignities the 

 entire character, withdrawing our thoughts from what is 

 mean and degrading, and inclining them to the noblest 

 ami worthiest of objects to the love and veneration, and 

 therefore to the practice, of TRUTH. 



These advantages, though unconnected with outward 

 and tangible results, are surely too precious to be 

 entirely overlooked in any correct estimate of the value 

 of scientific pursuits. When, in the course of these, you 

 find yourself engaged in an inquiry that may seem to 

 have but little relation to the things of sense, and are, 

 on this account, disposed to ask what is the use of it ( 

 just reflect whether the intellectual exercise has not been 

 combined with intellectual gratification ; and whether 

 there bo not an abstract beauty in the result arrived at, 

 that awakens pleasurable feelings, practically realised, 

 though hard to be described. We think the reflection 

 will, in general, suggest an answer to the inquiry in 

 some measure satisfactory. 



Those tranquil and purely intellectual enjoyments 

 have prompted and encouraged the efforts of the wisest 

 of men have cheered and sustained them amidst penury 

 ami neglect; and, under the persecutions of power, 

 wielded by ignorance, have supplied a consolation second 

 only to that which Divine revelation affords; and they 

 have finally been regarded as no moan reward for a life 

 thus tried, and toils thus endured. They were con- 

 sidered as compensation sufficient for the labours c 

 of NEWTON many of whose discoveries in science would, 

 in all probability, never have been given to the world 

 but for the urgent interference of private friends. 



\Ve are anxious that you should be influenced by con- 

 siderations such as these ; and that you should regard 

 science as something more than a ministering agent to 

 our animal comforts, or even to our social gratification 

 and convenience. The practical benefits of science and 

 more especially of those departments of it conn. 

 with the subjects of the present volume are in little 

 danger of being overlooked or undervalued; they are 

 spread profusely around us ; and are felt and enjoyed by 

 all. And there thus seemed to be all the more need for 

 directing your attention to collateral advantages less 

 palpable and striking, and therefore less likely to be 

 duly appreciated. 



We shall now proceed to the business before us : we 

 shall assume no knowledge at all on your part in re- 

 ference to the topics to bo discussed ; and, in even so 

 simple a subject as ARITHMETIC, we shall begin at the 

 beginning. 



Arithmttif. Arithmetic is that branch of knowledge 



