# Did Leibniz sketch a design for a machine capable of solving a system of linear equations?

My question is based on the information given in the book "the tangled origins of the leibnizian calculus", at pages 108 -109. I know that leibniz invented the stepped drum and used it to build the stepped reckoner - that first mechanical calculator capable of doing all four arithmetical operations. He also sketched a design for a different machine - the pinwheel calculator, designed a cipher machine, developed the binary arithmetic (based on 0 and 1) and established it's importance for computers, and even described in detail some of the fundamental principles of the modern computer (in his treatise "De progressione Dyadica"). With such achievements in hand, it's not an exageration to believe the statement in this book. So my question is, like many other questions of mine, can someone give a reference for leibniz's supposed machine? can someone give a reference for a drawing of leibniz machine (how does it look like?).

• I mean my conjecture about "a machine capable of solving a system of linear equations". All the other machines i mentioned were at least at the design stage (while the stepped reckoner was also built by him). – user2554 Jun 3 '16 at 13:52
• I come more and more to be convinced that a design of such complicated machine by leibniz is possible. Many websites state that leibniz designed far more complex machines than his stepped reckoner. – user2554 Jun 3 '16 at 18:59
• See also FLORIN-STEFAN MORAR, Reinventing machines: the transmission history of the Leibniz calculator (2015). – Mauro ALLEGRANZA Jun 19 '16 at 17:24
• Mauro ALLEGRANZA, is this article concerned with Leibniz's stepped reckoner or with his machine capable of solving equations (Constructeur Universel d'Equations) ? i ask because i dont need references for articles about the stepped reckoner (i know how the stepped reckoner worked). – user2554 Jun 20 '16 at 12:53

I've found an indirect reference into:

still in 1674 he [Leibniz] has plans for constructing an "analytical" counterpart to his calculating machine, an instrument for determining the solutions of equations, and he actually succeeds in doing so [footnote 27]

Footnote 27 for details see Hofmann (1970) ["Uber fruhe mathematische Studien von G.W.Leibniz", Studia Leibnitiana, 2, 81-114] : 101-4. The instrument is mentioned in letters to Oldenburg: June, July, Dec.1675; also in L-Huygens, mid-Sept.1675. From Huygens' rather cautious reply, 30 Sept.1675 we may conclude that he has seen a sketch of the "Compas d'equation", as had Tschirnhaus according to Leibniz's letter, 8 Jan.1694. [...] A similar instrument is described in the Encyclopédie, Suppl.II (1776): 834-5 (with an illustration in the Planche suppl.(1777)) under "Algèbre". This instrument works for quadratic equations only, but it is justly claimed as a prototype capable of adaptation to higher equations. The inventor is not named: he might indeed have been inspired by the hint in G.W.Leibniz, Der Briefwechsel mit Mathematikern I (Berlin 1899): 146 since the passage was available in print, in John Wallis, Opera mathematica III (Oxford 1699): 620-2 where the letter in question is reproduced, and in Commercium Epistolicum (1712): 45 where an excerpt is given [...].

• So certainly Leibniz did design such a machine. That's already an advance (before your answer i wasn't sure if the statement in the book is correct). I'm unable to read the references you gave since i dont know german or french, so i'll ask only if you have a reference in english or at least a drawing of the machine. – user2554 Jun 4 '16 at 16:02
• @user2554 - I've added the link to the Planche in the Suppl (1777) to the Encyclopédie, illustrating the Constructeur Universel d'Equations. – Mauro ALLEGRANZA Jun 4 '16 at 16:18
• Thanks you very much. The machine isn't similar to what i imagined (i imagined an enourmous machine with huge number of gears), so i guess it's probably more ingenius. It's regretable that so little of leibniz's technical writings were translated to english - they might have an extraordinary positive impact on people who want to trace back technological ideas and understand, for example, the history of mechanical computers. – user2554 Jun 4 '16 at 16:57