8 OBJECTS, ADVANTAGES, AND 



discovering- what they are ; or, suppose we would examine properties 

 belonging to all numbers ; this must be performed by a peculiar kind of 

 arithmetic, called universal arithmetic, or Algebra.* The common 

 arithmetic, you will presently perceive, carries the seeds of this most 

 important science in its bosom. Thus, suppose we inquire what is the 

 number which multiplied by 5 makes 10 ? this is found if we divide 10 

 by 5 it is 2 ; but suppose that, before finding this number 2, and 

 before knowing what it is, we would add it, whatever it may turn out, 

 to some other number ; this can only be done by putting some mark, 

 such as a letter of the alphabet, to stand for the unknown number, 

 and adding that letter as if it were a known number. Thus, 

 suppose we want to find two numbers, which, added together, make 

 9, and multiplied by one another make 20. There are many, which 

 added together, make 9 ; as 1 and 8 ; 2 and 7 ; 3 and 6 ; and so on. 

 We have, therefore, occasion to use the second condition, that multiplied 

 by one another they should make 20, and to work upon this condition 

 before we have discovered the particular numbers. We must, therefore, 

 suppose the numbers to be found, and put letters for them, and by rea- 

 soning upon those letters, according to both the two conditions of 

 adding and multiplying, we find what they must each of them be in 

 numbers, in order to fulfil or answer the conditions. Algebra teaches 

 the rules for conducting this reasoning, and obtaining this result success- 

 fully ; and by means of it we are enabled to find out numbers which are 

 unknown, and of which we only know that they stand in certain rela- 

 tions to known numbers, or to one another. The instance now taken 

 is an easy one ; and you could, by considering the question a little, 

 answer it readily enough ; that is, by trying different numbers, and 

 seeing which suited the conditions ; for you plainly see that 5 and 4 

 are the two numbers sought ; but you see this by no certain or general 

 rule applicable to all cases, and therefore you never could work more 

 difficult questions in the same way ; and even questions of a moderate 

 degree of difficulty would take an endless number of trials or guesses 

 to answer. Thus, if a ship, say a smuggler, is sailing at the rate of 

 8 miles an hour, and a revenue cutter, sailing at the rate of 10 miles an 

 hour, descries her 18 miles off, and gives chase, and you want to know 

 in what time the smuggler will be overtaken, and how many miles she 

 will have sailed before being overtaken ; this, which is one of the sim- 

 plest questions in algebra, would take you a long time, almost as long 

 as the chase, to come at by mere trial and guessing (the chase would 

 be 9 hours, and the smuggler would sail 72 miles :) and questions only 

 a little more difficult than this, never could be answered by any number 

 of guesses; yet questions infinitely more difficult can easily be solved 

 by the rules of algebra. In like manner, by arithmetic you can tell the 

 properties of particular numbers; as, for instance, that the number 348 

 is divided by 3 exactly, so as leave nothing over : but algebra teaches 

 us that it is only one of an infinite variety of numbers, all divisible by 

 3, and any one of which you can tell the moment you see it ; for they 

 *H have the remarkable property, that if you add together the figures 

 Jiey consist of, the sum total is divisible by 3. You can easily perceive 



* Algebra, from the Arabic words signifying the reduction of fractions; the Arabs 

 Having brought the knowledge ol it into Europe. 



