ABSTRACT
Let us quote Gauss, p. 292, Sec. 266
Thus far we have restricted our discussion to functions of the second degree with two unknowns and there was no need to give them a special name. But manifestly this argument is only one section of the general treatise concerning rational algebraic functions which are integral and homogeneous in many unknowns and many dimensions. Such functions relative to the number of dimensions can properly be divided into forms of the second, third, fourth degree, etc. and relative to the number of unknowns into binary, ternary, quaternary, etc. forms. Thus the forms we have been considering can be called simply binary forms of the second degree. But functions like Ax2+2Bxy+Cy2+2Dxz+2Eyz+Fz2https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003071808/b8e77471-25aa-4be1-82c4-566e277d4128/content/eq994.tif"/>
where (A, B, C, D, E, F are integers) are called ternary forms of the second degree and so forth. We have devoted the present section to the treatment of binary forms of the second degree. But there are many beautiful truths concerning these forms which are properly considered in the theory of ternary forms of the second degree. We will therefore make a brief digression into this theory and will especially treat of those elements which are necessary to complete the theory of binary forms, hoping thereby to please geometers who would be disappointed if we ignore them or treated them in a less natural manner. We must, however, reserve a more exact treatment of this important subject for another occasion because its usefulness far exceeds the limits of this work and because, hopefully, we will be able to enrich the discussion by more profound insights later on. At this time we will completely exclude from the discussion 76quaternary, quinary, etc. forms and all forms of higher degrees.* It is sufficient to draw this broad field to the attention of geometers. There is ample material for the exercise of their genius, and transcendental arithmetic will surely benefit by their efforts.
