GLEANINGS 
> 
IN 
SCIENCE 
No. 30. — June, 1831. 
I . — On Conjugate Hyperbolas. 
To the Editor of Gleanings in Science. 
Sir, 
It is an old remark, that scientific men are fonder of extending the boundaries of 
the department to which they are attached, than of examining its principles ; and 
that, in consequence, oversights are occasionally found to exist in the very outset 
of those sciences which have been most laboriously cultivated. 
What I am about to lay before you, will afford a striking illustration of the 
above observation. Few branches of science have been more assiduously cul- 
tivated than that of Conic Sections. They have been treated, as it should seem, in 
every variety of way, both with and without reference to the cone ; and yet it 
is a singular circumstance, that in all the treatises which I have had an oppor- 
tunity of examining, there is either no direction given for such an elementary 
proposition as that of cutting a pair of Conjugate Hyperbolas from Conjugate 
Cones, or the directions, if given, are erroneous. 
The books which I have consulted, are Robert Simson’s five books ; the trea- 
tise on Conics contained in the Scholia to the 8th Prop, book I. of Newton’s 
Principia, by Le Seur and Jacquier; the articles in Rees’ Encyclopaedia ; Abram 
Robertson’s (of Oxford) four books ; and the very recent work of H. P. Hamilton, 
of Cambridge, 1828. To these I may add, the little treatises of Vince and Peacock. 
Though all these Authors are diffuse in explaining the properties of conjugate 
hyperbolas, not one of them gives the least hint as to how conjugate hyperbolas 
can be cut from conjugate cones. 
The only author within my reach, who speaks expressly on this subject, is Dr. 
Hutton; and the following is his definition of conjugate hyperbolas, as given among 
the definitions to the chapter on Conic Sections, in the 2d volume of his course of 
Mathematics. “And further, if there be four cones, (PI. XIII. fig. 1.) CMN, COP, 
CMP, CNO, having all the same vertex C,' and all their axes in the same plane, 
and their sides touching or coinciding in the common intersecting lines MCO, NCP ; 
then if these four cones be all cut by one plane, parallel to the common plane of 
their axis, there will be formed the four hyperbolas, GQR, FST, VKL, WHI, of 
which each two opposites are equal, and the other two are conjugates to them ; as 
here, in figure 1.’’ To be sure of not misapprehending the meaning of the above 
definition, I have consulted all the mathematicians of my acquaintance, and they, 
NEW SERIES, NO. VI. 
