our purpose, and if, according to what has been taught, we substitute 1+y for z, we shall arrive at another biquadratic formula equally deceptive. In fact, this is one of those embarrassing formulas which often mock our ingenuity; nothing, therefore, remains for us to do, but to prove the impossibility of the problem proposed. We succeed better in attempting to assign x such that x2+2, and x2+11 may both be squares, and we find x=1, we may easily find as many other values as we please. 35. There is also a particular and useful method of determining x such that a2x2+bx+c,' and d2x2+ ex+f may both be squares, which we must not omit. We should first, when necessary, reduce by multiplication or division, the square coefficients of x2 to an equality, which is sufficiently easy; the operation afterwards will be plain from an example. To find such that x2-x+7, and x2-7x+1 may both be squares. Suppose x2-x+7=A2, and x2−7x+i= B2; whence, by subtraction 6x+6=A-B2, or (2x +2)×3=(A+B)×(A−B): now assume 2x+2=A +B, 3=A−B, whence x+3=A, and x − 1=B. Each of these will give the same value of x; if we use A, we have x2-x+7=x2+5x+2, from which we obtain x=, a value which will make both the formulas squares. 36. Let us find x such that ax+b, cx+d, and ex +f may all be squares. Put ax+b=y, whence x= and this value of x a aey2+a2ƒ—abe a2 : the question therefore is reduced to the finding of x such that the two quadratic formulas acy2+a2d—abc, and aey2+a2f-abe may both be squares. By applying this method to the formulas 1-x, 2 -x, 8-x, we find x=. =• 37. Such are some of the most general methods of resolving formulas involving only one unknown quantity; but by far the most difficult and curious part of my subject yet remains, I mean the solution of Diophantine problems, in which two or more numbers are required. On this inquiry I cannot enter at present, for my paper has already swelled beyond its intended limits, I must, therefore, defer the farther prosecution of this research to another opportunity. A List of the Subscribers to the first Volume of the Mathematical Correspondent. Science has civilized man: its noble and generous patrons ought always to be publicly known and gratefully remembered. G. B. Dr. Samuel Ackerly, New-York, Simon Ackerman, ditto, Samuel Adams, ditto, Francis Adincourt, Lansingburgh, New-York, Copies. 1 1 Mr. Agnew, New-Brunswick, New-Jersey, Joshua Alder, New-York, David Allinson, Burlington, New-Jersey, American Philosophical Society, Philadelphia, James Baehr, ditto, Henry Bates, ditto, Wheeler Barnes, Burlington, Vermont, 1 1 6 1 i 1 Abner Bennedict, Springfield, New-Jersey, Thomas Bennet, New-York, 1 1 Daniel Berry, ditto, Professor Blackburn, College of William and Mary, Virginia, 1 Drayton Blackwell, New-York, L. S. Burling, New-York, Everart Bush, ditto, Michael Burnham, ditto, Matthias Burnet, Norwalk, Connecticut, Jarvis Buxton, Newbern, North-Carolina, Joseph Caldwell, Chapel Hill, North-Carolina, Patrick Calliley, New-York, Geradus Camp, ditto, Malcolm Campbell, ditto, John Capp, Harrisburg, Pennsylvania, R. Cary, Staten-Island, New-York, James B. Carney, Newbern, North-Carolina, Philip Carney, New-York, George Carouthers, York-Town, Pennsylvania, William Carver, New-York, Capt. Uriah O. Champlin, ditto, George Chilton, Newark, New-Jersey, E. Childs, ditto, Clement C. Clark, New-York, Jacob Concklin, Rockland, New-Jersey, George W. Cook, Newbern, North-Carolina, John Coope, Chester County, Pennsylvania, ter S. Cortilyou, Long-Island, ward Cotton, Boston, ph Cowing, Alexandria, 1 Craggs, near Richmond, Virginia, 1 John D. Craig, Baltimore, James Gregan, Brooklyn, New-York, Henry Cullen, ditto, G. Davis, Washington, Missisippi Territory, William Cumberland, ditto, Mr. Cutting, ditto, James Dean, Montpelier, Vermont, Peter Deitrich, New-York, Abraham De Baum, New-York, Charles De France, Washington, Missisippi Territory, Daniel Dominick, ditto, 1 William Elliot, New-Brunswick, New-Jersey, 1 Restore Estlack, Woodstown, New-Jersey, 1. Mr. Gautier, New-York, John Garnet, New-Brunswick, New-Jersey, William Gibbons, Pennsylvania, Edward Giddons, Baltimore, Walter Gilbert, Washington, Missisippi Territory, James Gosman, ditto, 1 1 1 1 1 Peter M'Gowan, New-York, Timothy Greene, ditto, Daniel Grey, ditto, James Grey, Philadelphia, John Griscomb, Burlington New-Jersey, Daniel Halstead, ditto, Alexander Hamilton, ditto, T. Hamilton, Philadelphia, Thomas Hamersly, New-York, John M. Hanchel, Burlington, New-Jersey, James Hardy, New-York, Jonathan Harned, ditto, Thomas Harrison, ditto, John Henry, Baltimore, Adrian B. Hicks, New-York, Dr. Hicks, ditto, Thomas Holiday, Hackensack, New-Jersey, Mr. Hollingsworth, Fredericksburgh, Virginia, James Hornet, New-York, Jesse Hoyt, ditto, James Hughes, Philadelphia, Rev. Thomas P. Irving, Newbern, North-Carolina, |