Algebra

The greater number of his papers during this time were, however, contributed to the Berlin Academy. Several of them deal with questions on algebra. In particular:

• His discussion of the solution in integers of indeterminate quadratics, 1769, and generally of indeterminate equations, 1770.
• His tract on the theory of elimination, 1770.
• His memoirs on the general process for solving an algebraic equation of any degree, 1770 and 1771; this method fails for equations of an order above the fourth, because it then involves the solution of an equation of higher dimensions than the one proposed, but it gives all the solutions of his predecessors as modifications of a single principle.
• The complete solution of a binomial equation of any degree; this is contained in the memoirs last mentioned.
• Lastly, in 1773, his treatment of determinants of the second and third order, and of invariants.

Theory of numbers

Several of his early papers also deal with questions connected with the neglected but singularly fascinating subject of the theory of numbers. Among these are the following:

• His proof of the theorem that every integer which is not a square can be expressed as the sum of two, three, or four integral squares, 1770.
• His proof of Wilson's theorem that if n is a prime, then (n - 1)! + 1 is always a multiple of n, 1771.
• His memoirs of 1773, 1775, and 1777, which give the demonstrations of several results enunciated by Fermat, and not previously proved.
• And, lastly, his method for determining the factors of numbers of the form
  x² + ay².

Miscellaneous

There are also numerous articles on various points of analytical geometry. In two of them, written rather later, in 1792 and 1793, he reduced the equations of the quadrics (or conicoids) to their canonical forms.

During the years from 1772 to 1785 he contributed a long series of memoirs which created the science of differential equations, at any rate as far as partial differential equations are concerned. I do not think that any previous writer had done anything beyond considering equations of some particular form. A large part of these results were collected in the second edition of Euler's integral calculus which was published in 1794.

Lagrange's papers on mechanics require no separate mention here as the results arrived at are embodied in the Mechanique analytique which is described below.

Astronomy

Lastly, there are numerous memoirs on problems in astronomy. Of these the most important are the following:

• Attempting to solving the three-body problem results in the discovery of Lagrangian_points, 1772
• On the attraction of ellipsoids, 1773: this is founded on Maclaurin's work.
• On the secular equation of the Moon, 1773; also noticeable for the earliest introduction of the idea of the potential. The potential of a body at any point is the sum of the mass of every element of the body when divided by its distance from the point. Lagrange showed that if the potential of a body at an external point were known, the attraction in any direction could be at once found. The theory of the potential was elaborated in a paper sent to Berlin in 1777.
• On the motion of the nodes of a planet's orbit, 1774.
• On the stability of the planetary orbits, 1776.
• Two memoirs in which the method of determining the orbit of a comet from three observations is completely worked out, 1778 and 1783: this has not indeed proved practically available, but his system of calculating the perturbations by means of mechanical quadratures has formed the basis of most subsequent researches on the subject.
• His determination of the secular and periodic variations of the elements of the planets, 1781-1784: the upper limits assigned for these agree closedly with those obtained later by Le Verrier, and Lagrange proceeded as far as the knowledge then possessed of the masses of the planets permitted.
• Three memoirs on the method of interpolation, 1783, 1792 and 1793: the part of finite differences dealing therewith is now in the same stage as that in which Lagrange left it.