SECTION VII. 



MATHEMATICS. 



CHAPTER I. 

 ARITHMETIC. 



Is the previous pages, the attention of our readers has 

 been chiefly directed to the forces which operate on every 

 kind of matter, and to the different qualities of those 

 substances of which material bodies are formed. We 

 have hitherto scarcely required more than the ordinary 

 rules of arithmetic in the calculations of quantities, 

 time, and space ; but we shall now proceed to an entirely 

 different range of subjects. These will afford a full expo- 

 sition of the different branches of mathematical science, 

 the rules of which enable us to express the relative value 

 of both the forces acting on, and the quantities of, matter, 

 together with every other particular involving the use of 

 cumbers, symbols, itc. As each subject will have a 

 special introduction, it is not necessary that we should 

 here occupy ai>y considerable space in discussing their 

 nature. We shall rather confine ourselves to advising 

 our readers as to the best method of studying. We may 

 first impress on their minds the necessity of proceeding 

 ftep by ttep. If any person were to glance at a page in 

 trigonometry, or the differential calculus, it would be a 

 matter of no surprise if he were to hopelessly despair of 

 ever mastering the subject. On the other hand, an 

 acquaintance with the common rules of arithmetic and 

 algebra, and the elements of geometry having been once 

 acquired, little or no difficulty will be experienced in the 

 investigation of the higher branches of those subjects, 

 provided the student makes himself thoroughly ac- 



?uainted with every rule, $uccessively, as he proceeds. 

 t is incumbent, therefore, that the old Greek motto, 

 " Hasten slowly," should become the rule of every tyro in 

 mathematical science, if success and proficiency are to 

 be obtained. We must add, that the entire work, both 

 in the preceding and succeeding pages, has been so 

 arranged as to gradually inure the mind to close reasoning. 

 When we have to investigate the laws of astronomical 

 science, all the rules of each branch of mathematics will be 

 involved; and it is therefore essential to those who would 

 penetrate into the depths of that science, that they should 

 have become tolerably proficient in each branch of mathe- 

 matics. Without this, the study of astronomy is little 

 better than "star-gazing;" which, while it may afford 

 excitement to our imagination, has but little effect in 

 increasing our knowledge and expanding our mental 

 powers. 



Many of our readers may have no desire to become 

 deep mathematicians. On such we would urge that 

 mathematics may not only be studied for the sake of 

 their applications, but may be used as a mental gym- 

 nasium. In every other branch of science, truth is 

 veiled under theories ; whilst, in mathematics, she appears 

 unclouded by a single doubt. Each proposition of 

 geometry is absolutely demonstrable. No doubt can be 

 left of their truth ; and in such investigations the mind 

 undergoes an exact although severe training, which fits 

 it for exercise in the daily affairs of life, as much as for 

 the accurate study of nature and nature's laws. 



The articles on each of the following subjects have been 

 written by different authors, whose names have been 

 already mentioned in the table of contents. The editor 

 has ventured on such alterations in the original papers as 

 he deemed would assist the reader or student iu cases of 



apparent difficulty, which often occur to those unused 

 to the phraseology and technicalities of mathematical 

 science. The arrangement of each paper has been made 

 so that the preceding shall be introductory to those 

 which follow ; and thus the student may progress from 

 one subject to another, until he has become fully ac- 

 quainted with each branch of the science. 



Introductory. The present section is to be devoted to 

 that branch of study implied in the term MATHEMATICS 

 a term which comprehends one of the most extensive and 

 important departments of human knowledge. By most 

 people, it is considered also as one of the most difficult 

 departments ; and many, with time and talents for the 

 task, are deterred from entering upon a study which 

 would amply repay the expenditure of both, by this 

 mistaken prejudice. Every science, no doubt, has its 

 hard and knotty points ; and iu no intellectual pursuit 

 can distinction be attained without labour, thought, and 

 perseverance ; yet if there be one subject of scientific 

 inquiry which, more than any other, is distinguished by 

 the simplicity, certainty, and obviousness of its funda- 

 mental principles by the irresistible evidence by which 

 position after position is established and by the sys- 

 tematic gradations by which layer after layer of the 

 intellectual structure is completed that subject is 

 Mathematics. 



In other topics of research, there is generally more or 

 less of hypothesis or conjecture: there are obscure recesses, 

 into which the light of truth and demonstration cannot 

 penetrate, and where fancy and imagination are some- 

 times permitted to guide our steps. But there are no 

 perplexities of this kind in mathematics no ingenious 

 theories to mislead, and no conflicting opinions to be- 

 wilder ; our progress here is exclusively under the un- 

 erring direction of TROTH herself ; and it is her torch 

 alone that lights up the path. 



Whatever, therefore, may be the difficulties connected 

 with the study of mathematics, it is plain that they do 

 not arise from our having to grope our way in darkness 

 and uncertainty ; the asperities of the road are as clearly 

 revealed before us as the level and unobstructed track ; 

 and all that the earnest student requires, is some friendly 

 hand to aid him in surmounting these in the earlier 

 stages of his progress. 



It is this sort of aid that we here propose to supply. 

 We do not undertake to conduct the scientific inquirer 

 through the entire regions of mathematical research 

 ours is a far less ambitions aim : we write for the young 

 for the self-dependent the solitary and, perchance, 

 the unfriended student. The office we here take upon 

 ourselves will be performed, if we succeed iu the en- 

 deavour to assist him. This is the only object at which 

 we now aim ; and we think it right thus explicitly to 

 declare it, in order to forewarn those who may desire 

 information on the more recondite researches of science. 



It may be proper to mention, however, that although 

 we now propose to limit our labours to an exposition of 

 the elementary principles of mathematical learning, and 

 to economise space as much as possible ; yet, within the 

 bounds prescribed, we shall take care that every subject 



