Did Leibniz design a machine capable of solving algebraic equations?

Question

My question is based on the information on pages 108-109 of the book The Tangled Origins of the Leibnizian Calculus. I know that Leibniz

• invented the stepped drum and used it to build the stepped reckoner — the first mechanical calculator capable of doing all four arithmetical operations.

• sketched a design for a different machine‚ the pinwheel calculator.

• designed a cipher machine.

• developed binary arithmetic (based on $0$ and $1$) and established its 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.

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?)?

Answer 1

I've found an indirect reference into:

• Joseph Ehrenfried Hofmann, Leibniz in Paris 1672-1676 (1974, German ed.1949), page 30 and footnote 27:

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 [...].

Answer 2

I accidentally found a new source that partially discusses this machine of Leibniz. This source is available on this link , and includes several original drawings from Leibniz's Nachlass.

First of all, Leibniz termed it "Constructor, Instrumentum Algebraicum" in a 9 pages manuscrupt, and later wrote another manuscript (with 12 pages) with further elaborations of this idea. From an historical perspective, his machine belongs to the analog computing tradition (it is called a "Leibnizian analog computer" in the link I gave).

Here is a quote from this site:

The further development of the slide rule is the analog calculator. It became widespread in the 19th century and initially appeared in mechanical form and special types such as the planimeter, integraph, tide calculator or differential analyzer. After the Second World War, flexible electrical models were developed that were in use until the 1970s. They initially contained electron tubes and later transistors, and they could be programmed by plugging in cables and setting rotary switches. Three hundred years earlier, Leibniz had already described the concept of a specialised mechanical analogue calculator - probably the first in the history of mathematics - in two manuscripts dated December 1674, i.e. during his time in Paris.

Regarding the mechanical components of this machine, the following quote gives some ideas:

The Leibniz Constructor consists of rods, strings, pulleys and joints and is designed to find solutions to an algebraic equation. For example, it is designed to determine $x$ from $ax + bx^2 + cx^3 = d$. To do this, you first set the values ​​for a, b and c on the device and an arbitrary starting value $x_1$ for $x$. These settings change the rods and mean that you can read off a line of length $ax_1 + bx_1^2 + cx_1^3$ on the Constructor. This is probably not yet equal to d. By further moving and twisting the rods, the variable size $x$ can be changed so that $ax + bx^2 + cx^3$ becomes equal to $d$ within the accuracy limit, which would achieve the goal.

Later, Johann Andreas von Segner (1704-1777) invented a similar machine independently of Leibniz (whose work he could not know), and several years later the Englishman John Rowning presented another working model with further improvements. Rowning's model (1770) is the one presented in one of the links in Mauro ALLEGRANZA's answer.