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A1UTHMETIC.

Greeks and Romans would have brought the science of arithme-tic to a much greater degree of perfection, than they ever did,had they hit upon the method of expressing numbers by ten dis-tinct characters such as are now in use. But the idea of a ci pher,which may be stiled the key-stone of arithmetic, seems never tohave entered their minds j and thus, though they reckoned pro-perly enough by tens, yet not having characters proportionate toexpress their numbers, they involved their arithmetic in a laby-rinth of confusion, from which, for want of this clue, neither anKuclid, nor an Archimedes , with all their wonderful abilities, wereable to extricate it.

Numeration is the reading of any number so as to express tirejust value of its distinct parts or periods. Numbers, in order thatthey may be more easily read, when expressed by figures, are di-vided from the right band towards the leit, into periods and halfperiods, each half period consisting of three figures; the commonname of the first period being units, or ones ; of the second, mil-lions ; of the third, billions ; of the fourth, trillions, &c. Alsothe first half of any period is so many ones of it, but the latter halfis so many thousands of it. The following example exhibits asummary of this whole doctrine, and may be extended to sextil-lions, septillions, octillions, nouillions, &c. ad infinitum.

Quintillions. Quadrillions. Trillions.

Billions. Millions. Units.

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a

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A number expressing a quantity of one name or denomination, iscalled a simple number, as 20 pounds, or 17 gallons, or 5 days;and that representing a quantity of several names, is called a com-pound number, as 13 pounds 5 shillings and 6 pence, or 17 gallonsand 2 pints, or 3 hours and 50 minutes.

(9.) To express in Figures any Number proposed inWords.

Write down cyphers or other marks to as many places as arenamed in the given number, under these marks write the signifi-cant figures according to their value, and fill up the remainingplaces with cyphers Sixty thousand two hundred and six is ex-pressed thus: 602 06', the dots being of no further use than to as-sist the tyro in adjusting the places of the figures.

(10.) To read or express in AVords any Number

EXPRESSED IN FIGURES.

Divide the figures in the given number, as in the general ex-ample above, into periods and half periods, by any convenientmarks; then beginning at the left, the figures are thus read, viz.the first figure of each half period is named by itself with the wordhundreds, but the other two are named together; and at the endof the first half of each period, the word thousands is named ; butat the end of the other half, the common name of the whole pe-riod, except it be the units period, the term units never being ex-pressed, but understood when it occurs in reading any number.

Thus 605003070600004 may have the periods and half periods

marked by double and single dots or otherwise, and then is easilyread as follows : six hundred and five billions three thousand andseventy millions six hundred thousand and four.

Sim pi e Addition.

(11.) Simple Addition is the finding of one siinple numberequal to several simple numbers taken all together. The numberwhich is equal to several taken together is called their sum.

Rule I. Place the several numbers, to be added, so that thefigures of the same name, with respect to units, tens, &c. maybe directly under each other.

2. Draw a line under the lowest number; then add up thecolumn of units, and beneath the line directly under the addedcolumn, write down the excess above the tens in the sum, andfor the tens carry as many ones to the next column.

3. Add all the columns in the same manner, and the figuresbelow the line will express the sum required.

Proof. Add up all the columns as before beginning at theupper end, and proceeding in a contrary order.

Example.

2591

45927

621

13243

7931

Sum 70318

Simple Subtraction.

(12.) Siinple subtraction is the finding how much one simplenumber exceeds another, or the taking a less simple number outof a greater. The number to be subtracted is called the sub-trahend ; and that out of which it is to be taken the minuend ;also the number remaining after the subtraction, is named theirdifference.

Rule 1 . Place the subtrahend under the minuend accordingto the directions given in addition, and draw a line below them.

2. Begin at the right, and subtract each under-figure from thatwhich stands above it, writing the remainder straight under thembelow the line; so shall all the remainders together express tintdifference required.

3. But when any under figure exceeds that which is above it,conceive 10 to be added to the upper, and subtract the underfrom the sum ; but in this case, you must add 1 to the next underfigure, before you subtract it.

Proof. Add the difference and subtrahend together, and thesum will be equal to the minuend when the operation is right,

Example. Minuend, 159327Subtrahend, 61489

Difference, 97838

Proof, 159327Simple Multiplication.

(13.) Simple multiplication is the finding of the simple number,which is produced from any two given simple numbers when oneof them is repeated as often as there are units in the other ; and itis therefore a compendious method of addition.

The two proposed numbers are in general, termed the factors ofthe multiplication; but in particular, that which is to be multi-plied, is called the multiplicand; and that by which it is multi-plied, the multiplier; the number found from the operation isnamed the product of the two factors.

The following table which contains the products of every two ofthe nine digits must be committed to memory before any opera-tions are performed.

Multiplication