A COURSE

OF

MATHEMATICS.

PRELIMINARY DEFINITIONS.

MATHEMATICS is the science which treats of all quantities that can be numbered or measured. Its two great divisions are pure mathematics and mixed mathematics.

Pure mathematics consists of the three following divisions:

1. Arithmetic, which treats of numbers or particular quantities; 2. Algebra, which treats of the relations of any quantities whatever under particular conditions, and may properly be termed Universal Arithmetic; and

3. Geometry, which treats of extended quantities, or continued magnitudes, as possessing three dimensions, viz., length, breadth, and thickness.

This last division embraces a much greater compass and variety of reasoning than either of the other divisions, and all of them are founded on the simplest notions of abstract quantities. The applications of these three divisions, one to another, form other important parts of pure mathematics.

Mixed mathematics is the application of the different parts of pure mathematics to those physical inquiries which are founded upon principles deduced from experiment or observation. It comprehends Mechanics, or the science of equilibrium and motion of bodies; Astronomy, in which the motions, distances, etc., of the celestial bodies are considered; Optics, or the theory of light, besides various other important subjects. In all these branches of mixed mathematics, if the first principles be accurately determined by experiment or observation, the results which are deduced are as certain and indisputable as those which can be deduced by geometry, or by any other part of pure mathematics, from axioms and definitions.

PRINCIPLES OF ARITHMETIC.

NUMERATION.

ART. 1. Arithmetic is that division of pure mathematics which treats of numbers, and of the method of performing calculations by means of them.

Number is a collection of several objects of the same kind, or of many

VOL. I.

B

separate parts. It is one of the forms of magnitude, an attribute or quality of objects by which they are conceived to be susceptible of increase or diminution. The other form of magnitude is distinguished by the connexion or continuity of the parts,--being an entire mass without distinction of parts; whereas in number the consideration is merely how many parts it contains. The definition of number supposes the existence of one of the things or parts of which it is composed, taken as a term of comparison, and which, in that case, is denominated unity.

2. Some knowledge of numbers must have existed in the earliest ages of the world. The ten fingers with which man had been formed, the flocks and herds which he had acquired, and the variety of objects that surrounded him, would all contribute to impress his mind with a notion of number. While small numbers only were required, the ten fingers would furnish the most convenient way of reckoning them, since with his fingers any person could make those little calculations which his limited wants required. He would name all the different collections of his fingers, and frame appropriate words, in his own language, answering to one, two, three, four, five, six, seven, eight, nine, and ten. As his wants increased he would proceed to higher numbers, adding one continually to the former collection, as he advanced from lower numbers to higher. He would soon perceive that there is no limit to the different numbers that may be formed, and consequently that it would be impossible to express them in ordinary language by distinct names independent of each other. By arranging numbers in groups or classes, they might be expressed by a comparatively small number of words, still the continual repetition which unavoidably occurs in calculation would necessarily preclude the use of names of numbers, except in operations of the very simplest character.

3. The English names of numbers have been formed from the Saxon language, by combining the names of the first ten numbers mentioned in the preceding article.

4. For facilitating calculations it would be found necessary to substitute short and expressive signs for words, and when some few signs or characters had been chosen, to combine them so as to represent the names of all other numbers whatever. We shall here show how this has been done by the Greeks and Romans, and then advert to the admirable system of notation which so generally prevails among different nations of the world at the present time.

5. The Greek Notation.-At a very early period the Greeks had recourse to the twenty-four letters of their alphabet for the representation of numbers, and by means of these, aided by the three Hebrew characters 5, 5., they expressed the first three orders of numbers. Thus the numbers one, two, three, etc., to nine, were represented by

α, B, y, ô, ε, 5, 5, n, e... (1st order, or units).

The numbers ten, twenty, thirty, etc., to ninety, by

1, K, λ, μ, v, 4, 0, π, 5

(2nd order, or tens);

and the numbers one hundred, two hundred, etc., to nine hundred, by
P, o, T, v, P, X, Y, w, x (3rd order, or hundreds).
Instead of multiplying distinct characters for higher numbers, they had
recourse to their characters for the units, and by subscribing a small iota
or dash, they denoted one thousand by a, two thousand by B1, and so
on. With these characters the Greeks could express every number
under ten thousand. Thus

050 signified nine thousand nine hundred and ninety-nine,
four thousand three hundred and eighty-two,
three thousand and one.

διτ πβ
για

In order to express higher numbers, they made use of the letter M, which, on being written below any character, increased its value ten thousand times. This contrivance enabled them to express all numbers as far as hundreds of millions; but instead of subscribing the letter M, it was afterwards found more convenient to write the letters Mv., or the contraction for pvpia, ten thousand, after the character,whose value was to be increased ten thousand times; and then, when lower periods failed, they repeated the letters Mv. Thus

λð Mu. Mu. Mv. signified thirty-four trillions.

Archimedes, the most inventive of the Greek philosophers, divided numbers into periods of eight symbols each, which were called octades ; and the famous Appollonius again divided them into periods of four symbols each, the first period on the left being units, the second myriads, the third double myriads, and so on. In this manner Appollonius was able to write any number which could be expressed by the present system of numeration. Having thus given a local value to his periods of four, it was remarkable that Appollonius did not perceive the advantage of making the period consist of a less number of characters. Had he done the same thing with every single character, he would have arrived at the system now in common use, and this oversight is the more remarkable as the cipher was not unknown to the Greeks, but confined exclusively to their sexagesimal operations.

6. The Notation of the Romans.-The traces or strokes which originally represented numbers were replaced by those characters of the Roman alphabet which most nearly resembled them. The Roman notation was much ruder than the Greek, and for the expression of number they made use of the seven following capital letters, viz. :

I for one; V for five; X for ten; L for fifty; C for a hundred; D for five hundred; and M for a thousand.

By various repetitions and combinations of these they expressed all numbers. The four combined strokes which originally formed the character M for a thousand, assumed afterwards a rounded shape, fre

quently expressed by the compound character CIO, consisting of the letter I inclosed on both sides by C, and by the same character reversed. This last form, by abbreviation on either side, gave two portions, one of which I was condensed into the letter D and expressed five hundred. The practice of using duodeviginti for octodecim, and so on, led the Romans to the application of deficient numbers; and instead of writing VIIII for nine, they counted one back from ten, and placing I before X, they wrote it thus, ÍX. In a similar manner XIX represented nineteen, XL forty, XC ninety, and CM nine hundred. They also repeated the symbols of a thousand to denote higher numbers; thus CCI represented ten thousand, and CCCIOƆƆ an hundred thousand. Separating each of these, gives IƆƆ for five thousand, and IƆƆƆ for fifty thousand. Also a horizontal line drawn over any letter augmented its value one thousand times; thus LX signified sixty thousand. With this explanation the following examples will be readily understood:

Thus as often as any symbol is repeated, its value is repeated as often ; a symbol of less value placed after one of greater value is added to the greater, but if placed before a symbol of greater value, it is subtracted from it. Also every added to IƆ increases its value ten times, and if a C be placed before CIO and a after it, its value is increased ten times, and so does every additional C and 3. This notation is still frequently employed in distinguishing dates, the chapters and sections of books, and so on.

7. Although the Greek arithmetic, as successively moulded by the ingenuity of Archimedes, of Appollonius, and of others, had attained to a high degree of perfection, and was capable of performing operations of very considerable difficulty and magnitude, still the great and radical defect of the system consisted in the entire absence of a general mark corresponding to our cipher, which without having any value in itself, should yet serve to keep the rank or power of the other characters, by occupying the vacant places in the scale of numeration. From the preceding remarks on the notation of the Greeks and Romans, the student will be able to form some idea of the great superiority of the present system, which has led to some of the most striking and remarkable scientific discoveries.

8. In the common system of numeration all numbers, however large or small, can be expressed by the ten following characters or figures, viz. :

1 2 3 4 5 6 7 8 9 0 one, two, three, four, five, six, seven, eight, nine, nothing. The first nine of these are called significant figures or digits, and sometimes represent units, sometimes tens, hundreds, or higher classes. When placed singly, they denote the simple numbers subjoined to the characters; when several are placed together, the first figure on the right is taken for its simple value, the next signifies so many tens, the third so many hundreds, and the others so many higher classes corresponding to the order in which they are placed. Thus 4532 signifies four thousand, five hundred, thirty, and two units; and in the number two hundred and twenty-two, which is written thus 222, the figure 2 is repeated thrice, but each has a different value; the first on the right hand is two units, the second two tens or twenty, and the third two hundreds.

9. When any of the denominations, units, tens, hundreds, etc., is wanting, it becomes necessary to supply its place with the last sign or character, viz., 0, which is termed cipher or nothing, the word cipher in the Arabic language signifying vacuity. This character which indicates the absence of all number, is a most important oue, inasmuch as its introduction serves to preserve the proper positions of the significant figures. Thus the number forty thousand three hundred and twenty would be expressed in figures by 40320, because the denominations, units, and thousands are wanting, and the absence of each is indicated by the cipher which occupies its place. From these illustrations we may perceive that the superiority of our present system of numeration arises from a few simple signs being made to change their value as they change the position in which they are placed, and that the significant figures have a local as well as a simple value. It is thus that, in consequence of the established relative value of units and tens, the same figure which, beginning on the right, expresses units, becomes ten times greater at each remove to the left, and by simply changing their places, the different characters become susceptible of representing successively all the different collections of units which can possibly enter into the expression of a number. Thus we get

10 11
ten, eleven, twelve, thirteen, fourteen, fifteen,

16

17

18

19

sixteen, seventeen, eighteen, nineteen ;

where the first figure on the left signifies ten, and the second figure its simple value, or so many units. Hence 10 means ten and nothing; 11 ten and one, and so on. Again, 20 means two tens and nothing, or twenty; 21 two tens and one, or twenty-one; 30, thirty; 90, ninety; 100, ten tens or one hundred; and 1000, one thousand. The names and values of numbers will be readily acquired from the following examples.