ARITHMETIC 527 Robert Recorde (died 1558), whose Grounde of Artes and Whetstone of Witte were arithmetical works of great value ; Nicolo Tartaglia (died 1559); Michael Stiefel or Stifelius (1486-1567), said to have been the inventor of the signs + and -; Peter Ramus (1515-1572); Albert Girard (died 1634); William Oughtred (died 1660); Edward Cocker, whose Arithmetic (1st ed. 1677), a posthumous work, and probably a forgery, is of slight merit, though it passed through many editions ; Kastner (Geschichte der Mathematik, 4 vols. 1796-1800); Montucla (Histoire des Jfathematiques, vol. i. 1799); De Morgan (Arithmetical Books, 1847). By far the best history of arithmetic is that uf Dr George Peacock, late Dean of Ely, published in the Encyclopaedia Metropolitana. Of recent works on arithmetic there is great abundance, and many of them are of great excellence. They usually contain statements of modes of operation, under the name of " Rules," with a number of examples under each for practice ; and not a few of them give in addition explana tions of the rationale of the methods. In the following sections the ordinary processes of arith metical calculation and their commonest practical uses are briefly explained. The various methods of operation are given with greater or less detail, as has appeared necessary for the exposition of the principles on which the operations depend ; and light is in many instances thrown on both processes and principles by illustrative examples. For further examples the reader is referred to the manuals above alluded to. The earlier sections, forming the greater part of the article, are occupied with numbers in the ab stract, and the remainder with arithmetic in its practical applications. I. ABSTRACT ARITHMETIC. 1. dotation is the name usually given to the expressing of numbers by means of characters or figures. The number ten is the basis or radix of the Arabic sys tem, of notation, and every number may be expressed in that notation by combinations of the ten digits, or numeral figures, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, the last (the cipher or zero) having no value except in combination. When several of these are placed together, every removal towards the left increases the value of a figure ten times. The figure placed furthest to the right has the same significance as when it stands alone, i.e., it represents units, the figure next to it denotes tens, the next hundreds, the next thou sands; 8735 is therefore read as eight thousand seven hundred and thirty-five. With more than four figures we should have still proceeding to the left tens of thou sands ; then hundreds of thousands, and then thousands of thousands; but instead of this we use the term millions. A million of millions is a billion, 1 and a million of billions a trillion. A digit followed by three figures thus expresses thousands; by six figures, millions; by twelve, billions; and so on. Beyond trillions we have taking the figures in groups of six quadrillions, quintillions, sextillions, sep- tillions, octillions, nonillions, etc. But occasion is very seldom found to employ these terms, and indeed it is diffi cult to form any distinct or definite idea of even a billion. The system of notation in ordinary use is remarkable at once for its simplicity and its completeness. The selection of ten as the basis or radix of the system (whence numbers expressed in this way are said to be in the denary scale of notation) is in all probability to be traced to primitive 1 In France, and by a few English writers, a thousand millions is called a billion, a thousand billions a trillion, c. The ordinary Eng lish usage is at least as convenient, and agrees (which the other does not) with the etymological formation of the words the billion, tril lion, quadrillion, &c., being respectively the second, third, fourth, &c., powers of a million. calculations by means of the ten fingers. Ten is well chosen, being neither too large nor too small, but twelve might, in some respects, have been found more convenient. All numbers can be expressed with twelve or any other number as basis, just as with ten. 2 2. Numeration is the art of reading figures employed to Numera- express numbers. The following table shows the places tion - of the figures, as already described, up to billions : (1.) Units. (2.) Tens. (3.) Hundreds. (4.) Thousands. (5.) Tens of thousands. (6.) Hundreds of thousands. (7.) Millions. (8.) Tens of millions. (9.) Hundreds of millions. (10.) Thousands of millions. (11.) Tens of thousands of millions. (12.) Hundreds of thousands of millions. (13.) Billions. The figures 8,607,034,740,952, for example, are read thus: Eight billions, six hundred and seven thousand and thirty-four millions, seven hundred and forty thousand, nine hundred and fifty-two. Ciphers are passed over in reading, their function being to determine the proper posi tion of the significant figures. When the number of figures exceeds four, it is usual to print them off as above in groups of three, by means of commas. At the odd commas, reckon ing from the right hand, we always read thousands ; at the even commas, millions, billions, <tc. 3. Notation has, in addition to the general meaning already explained, a special signification as the converse of numeration, i.e., it is the art of representing by figures numbers that are given expressed in words. 4. Addition is the method of finding the sum of two or Addition. more given numbers, that is, the number to which they amount when taken together. Suppose a column, added in the ordinary way, amounts to 34. The 4 is set down under the column, and the 3, representing 30, is added with (or, as usually expressed, " carried to ") the next column. 5. Subtraction is the method of finding the difference Subtrac- between two given numbers, that is, the number by which tion - the greater number exceeds the less, or the remainder after the less is taken from the greater. In subtracting, say, 38 from 92, since 8 cannot be taken from 2, we "borrow" from the place of tens, i.e., the 2 is increased by 10, and 8 taken from 12 leaves 4. The 10 added (or " borrowed ") has to be taken away again, and this is done by taking 1 from 9 in the place of tens. But 3 has also to be taken from this 9 ; so first of all the 1 and 3 are added together, i.e., 1 is "carried" to the 3, and 4 taken from 9 leaves 5, giving 54 as the complete re mainder. The " borrowing " process depends on the con sideration that when the same amount is added to both numbers their difference is unaltered it being remembered that ten in the upper line corresponds to one a place to wards the left in the lower line. 6. Multiplication is a method of finding the result pro- Multipli- duced by adding a given number taken a given number cation, of times. The number to be repeated is called the multiplicand; the number expressing the times it is taken, the multiplier; and the result obtained, the product. The 2 With twelve as basis (i.e., in the duodenary scale), the number represented by the figures 8735 would be the sum of 8 x 12 x 12 x 12, 7 x 12 x 12, 3 x 12, and 5, i.e., 14,873 in the ordinary or denary scale. To convert 8735 of the denary into the duodenary scale, we find the twelves in it, that is, divide it by 12, the quotient by 12, &c. This
gives (using the character e for eleven), 507e.