=1735.= All the fruitful uses of imaginaries, in Geometry, are those which begin and end with real quantities, and use imaginaries only for the intermediate steps. Now in all such cases, we have a real spatial interpretation at the beginning and end of our argument, where alone the spatial interpretation is important; in the intermediate links, we are dealing in purely algebraic manner with purely algebraic quantities, and may perform any operations which are algebraically permissible. If the quantities with which we end are capable of spatial interpretation, then, and only then, our results may be regarded as geometrical. To use geometrical language, in any other case, is only a convenient help to the imagination. To speak, for example, of projective properties which refer to the circular points, is a mere _memoria technica_ for purely algebraical properties; the circular points are not to be found in space, but only in the auxiliary quantities by which geometrical equations are transformed. That no contradictions arise from the geometrical interpretation of imaginaries is not wonderful; for they are interpreted solely by the rules of Algebra, which we may admit as valid in their interpretation to imaginaries. The perception of space being wholly absent, Algebra rules supreme, and no inconsistency can arise.--RUSSELL, BERTRAND. _Foundations of Geometry (Cambridge, 1897), p. 45._ =1736.= Indeed, if one understands by algebra the application of arithmetic operations to composite magnitudes of all kinds, whether they be rational or irrational number or space magnitudes, then the learned Brahmins of Hindostan are the true inventors of algebra.--HANKEL, HERMANN. _Geschichte der Mathematik im Altertum und Mittelalter (Leipzig, 1874), p. 195._ =1737.= It is remarkable to what extent Indian mathematics enters into the science of our time. Both the form and the spirit of the arithmetic and algebra of modern times are essentially Indian and not Grecian.--CAJORI, F. _History of Mathematics (New York, 1897), p. 100._ =1738.= There are many questions in this science [algebra] which learned men have to this time in vain attempted to solve; and they have stated some of these questions in their writings, to prove that this science contains difficulties, to silence those who pretend they find nothing in it above their ability, to warn mathematicians against undertaking to answer every question that may be proposed, and to excite men of genius to attempt their solution. Of these I have selected seven. 1. To divide 10 into two parts, such, that when each part is added to its square-root and the sums multiplied together, the product is equal to the supposed number. 2. What square is that, which being increased or diminished by 10, the sum and remainder are both square numbers?