Hence the signs plus and minus were considered, not as merely denoting the relation of one quantity to another placed before it, but, by a kind of fiction, they were considered. as denoting qualities inherent in the quantities to the names of which they were prefixed. This fiction was found extremely useful, and it was evident that no error could arise from it. It was necessary to have a rule for determining the sign belonging to a product, from the signs of the factors composing that product, independently of every other considera tion ; and this was precisely the purpose for which the above fiction was introduced. So necessary is this rule in the generalizations of algebra, that we meet with it in Diophantus, notwithstanding the imperfection of the language he employed ; for he states, that 4044; into e*4,c gives •Tirsegg, &c. The reduction, therefore, of the operations on quanti ty to an arithmetical form, necessarily involves this use of the signs plus or minus ; that is, their application to denote something like absolute qualities in the objects they collect together. The attempts to free algebra from this use of the signs have of course failed, and . must ever do so, if we would preserve to that science the extent and facility of its operations.
Even the most scrupulous purist in mathematical language must admit, that no real error is ever introduced by employing the signs in this most abstract sense. If the equation q.r +r=o,• be said to have one positive and two negative roots, this is certainly as exceptionable an application of the term negative, as any that can be proposed ; yet, in reality, it means nothing-but this intelligible and simple truth, that + + qa• r. b) (.v c); or that the former of these quantities is produced by the multiplica tion of the three binomial factors, .r—a, b, .r c. We might say the same nearly as to imaginary roots ; they show that the simple factors cannot be found, but that the qua dratic factors may be found ; and they also point out the means of discovering them.
The aptitude of these same signs to denote contrariety of position among geometric magnitudes, makes the foregoing application of them infinitely more extensive and more indispensable.
From the same source arises the great simplicity introduced into many of the theorems and rules of the mathematical sciences. Thus, the rule for finding the latitude of a place from the sun's meridian altitude, if we employ the signs plus and minus for indicating the position of the sun and of the place relatively to the equator, is enunciated in one simple proposition, which includes every case, without any thing either complex or ambiguous. But if this is not. done,—if the signs plus and minus are not employed, there must be at least two rules, one when the sun and place are on the same side of the equator, and another when they are on different sides. In the more complicated calculations of spheri
cal trigonometry, this holds still more remarkably. When one would accommodate such rules to those who are unacquainted with the use of the algebraic signs, they are perhaps not to be expressed in less than four, or even six different propositions ; whereas, if the use of these signs is supposed, the whole is comprehended in a single sentence.. In such cases, it is obvious that both the memory and understanding derive great advantage from the use . . of the signs, and profit by a simplification, which is the work entirely of the algebraic language, and cannot be imitated by any other.
That I might not interrupt the view of improvements so closely connected with one another, I have passed over one of the discoveries, which does the greatest honour to the seventeenth century, and which took place near the beginning of it.
As the accuracy of astronomical observation had been continually advancing, it was ne cessary that the correctness of trigonometrical calculation, and of course its difficulty, should advance in the same proportion. The signs and tangents of angles could not be ex . pressed with sufficient correctness without decimal fractions, extending to five or six places below unity, and when to three such numbers a fourth proportional was to be found, the work of multiplication and division became extremely. laborious. Accordingly, in .the end of the sixteenth century, the time and labour consumed in such calculations had become excessive, and were felt as extremely burdensome by the mathematicians and astronomers all over Europe. Napier of Merchiston, Whose mind seems ,to have been peculiarly turn ed to arithmetical researches, and who was also devoted to the study of astronomy, had early sought for the means of relieving himself and others from this difficulty. He had viewed the subject in a variety of lights, and a number of ingenious devices had occurred to him, by which the tediousness of arithmetical operations might, more or less completely, be avoided. In the course of these attempts, he did not fail to observe, that whenever the numbers to be multiplied or divided were terms of a geometrical progression, the product or the quotient must also be a term of that progression, and must occupy a place in it pointed out by the places of the given numbers, so that it might be found from mere in. spection, if the progression were far enough continued. If, for instance, the third term of the progression were to be multiplied by the seventh, the product must be the tenth, and if the twelfth were to be divided by the fourth, the quotient must be the eighth ; so that the multiplication and division of such terms was reduced to the addition and subtrac tion of the numbers which indicated their places in the progression.