Consul, The Educated Monkey

“The Educated Monkey” can not only count in a row (as it did in Mikhail Zoshchenko’s short story with the same title), but can multiply and add numbers! If you move the monkey’s “legs” to the given numbers, its “hands” fingers will point at the result of the chosen oper­a­tion.

The middle buttons switch the viewing mode between a histor­ical model, its kine­matic scheme or a combined view.

Slide foot to point
at the number desired
Slide foot to point
at the number desired
Fingers will locate the numbers' product

The history of creating an “Edu­cated monkey” is related to a trained monkey, named Consul, visit to United States in the early 20th century. It was able to imitate many human actions, including oper­ating the cash register. It partic­i­pated in many popular shows in different parts of the USA. In 1909 even a short film titled “Consul crosses the Atlantic” was released — at that time, the planes couldn’t fly long distances, one could get to America by ship only, spending more than a week in a voyage across the Atlantic ocean.

In 1910, the monkey performed in city of Dayton, where the NCR company, which produced cash regis­ters and held a large collec­tion of such devices, was situ­ated. (Now this company is known for its ATMs, which are also used in Russia).

In 1915, NCR’s draftsman William Henry Robertson, who previ­ously worked as a math­e­matics teacher in secondary school and had heard stories about Consul visits to the company, applied for two patents. The first patent covered “cal­cu­lating device” for a “quick and simple method of searching results” on the diagram. Such a method for solving equa­tions is called nomog­raphy. The second patent covered a toy, based on the same mech­a­nism, “for stim­u­lating chil­dren’s interest in learning numbers”. The mentioned toy was imple­mented as a monkey. Finally, in 1916 the toy “Consul: The Educated Monkey” appeared in cata­logues.

“The Educated Monkey” can multiply and square numbers from 1 to 12. The reason is that States had inher­ited “school” multi­pli­ca­tion table from Britain. In English system of length measures 1 foot is equal to 12 inches; until 1971 the mone­tary unit of 1 shilling was equal to 12 pence, and in weight system of measures, used by The Royal Mint to this day, 1 pound equals 12 ounces. The multi­pli­ca­tion table, being an educa­tion system instru­ment, is “adapted” to the country and in the US, it is tradi­tion­ally 12×12. In Russia metrical (decimal) measure system was imple­mented as a non-manda­tory system at the end of 19th century, and as a manda­tory — in 1917—1925. Thus, Russians are used to a 10×10 multi­pli­ca­tion table.

“The Educated Monkey” is a planar hinge mech­a­nism.

The construc­tion of the device is symmet­rical, the basis of the mech­a­nism is two equal isosceles right-angled trian­gles. Each hypotenuse at one end is resting on a number line, where these support points ($L$ and $R$) can slide. Another ends of trian­gles’ hypotenuses are hinged. (Such a construc­tion can be imag­ined as two halves of a square cut along a vertical diag­onal, the lower points of which can be slid apart).

Two addi­tional links, hinged to each other in a point O, are also hinged to trian­gles at the vertex of the right angles. These links are equal in length to trian­gles’ catheti and together with them form a hinged rhombus.

The point $O$ is always in the middle between points $L$ and R: just like the hinge connecting the trian­gles, it lies on perpen­dic­ular bisector of segment LR. The choice of isosceles right-angled trian­gles ensures linear (with a coef­fi­cient of $1/2$) depen­dence of the point $O$ height above the number line on the segment $LR$ length — distance from the point $O$ to the line is always half the length of the segment $LR$ (a strict proof is given below).

Since inte­gers on the number line are uniformly distrib­uted, then at all possible posi­tions of $L$ and $R$ corre­sponding to the inte­gers, the point $O$ posi­tions will form a triangle of values, whose rows will be evenly spaced verti­cally and the posi­tions inside one row will be in the middle between the posi­tions in adja­cent rows.

For example, if the points $L$ and $R$ are placed on the two adja­cent inte­gers, the point $O$ is located between them, at a distance of $1/2$ from the number line. If the points $L$ and $R$ are placed at a distance of two inte­gers, then the point $O$ is located at a distance 1 from the number line in the middle between the posi­tions in the previous row.

Thus, the “Edu­cated Monkey” imple­ments a binary oper­a­tion: each posi­tion of “vari­ables” $L$ and $R$ defines the unique posi­tion of the window, where the result of this oper­a­tion can be written. If the order of the numbers on which points $L$ and $R$ are placed is not consid­ered, the oper­a­tion will be commu­ta­tive i.e. it won’t depend on order of the numbers. For example, it’s possible to imple­ment func­tions of multi­pli­ca­tion and addi­tion. If we addi­tion­ally agree that when subtracting, the smaller number is always subtracted from the bigger one, and when dividing, the bigger number is always divided by the smaller, then it’s possible to imple­ment non-commu­ta­tive func­tions of subtrac­tion and divi­sion.

To apply the chosen oper­a­tion to two iden­tical numbers, for example, to get a square when multi­plying, a special extra “square” posi­tion is added for the point $R$ to the right of the number line. The triangle of values is expanded with corre­sponding numbers — squares of inte­gers where the $L$ point is placed.

For each chosen oper­a­tion a specific triangle of values is composed and placed on the field.

For a complete descrip­tion of the classic model of “The Educated Monkey” it’s worth noting that the rectan­gular window, which is pointed at by the monkey’s fingers, is placed above the point $O$. To ensure that the window is always in upright posi­tion, it’s made as a part of “tail” link, which slides its slot on the hinge connecting the trian­gles, located in a butterfly. The same link also holds the head in an upright posi­tion.

The “tail” link isn’t needed to make a simpli­fied model without the head and with round window right at the $O$ point. This simpli­fi­ca­tion also makes it so the “tail” does not protrude beyond the field even at distant posi­tions of the $L$ and $R$ points.

Let’s prove that the choice of main blue trian­gles as isosceles right-angled in the construc­tion of the monkey ensures linear (with a coef­fi­cient of $1/2$) depen­dence of the point height above the number line on the segment length. It is required so the rows of values triangle are evenly spaced verti­cally.

To find the height of the $O$ point let us circum­scribe a rectangle with hori­zontal and vertical sides around one of the isosceles right-angled trian­gles. One vertical side of a rectan­gular lies on symmetry axis of the construc­tion, while the other goes through the vertex of the blue triangle’s right angle. The resulted red-colored trian­gles are equal (they are right-angled, their catheti are equal and their angles are also equal). Let’s draw a hori­zontal diag­onal of a green hinged rhombus, a hori­zontal segment through the point $O$ and a vertical segment through the point $R$. Three red segments are equal. Since the hori­zontal diag­onal of rhombus divides the vertical diag­onal in half, the pink segment has the same length. Thus, the violet segments are also equal. There­fore, the violet quadri­lat­eral is always — in every posi­tion of the $L$ and $R$ points — a square. One of its sides is equal to the height of the $O$ point above a number line, and the other, because of symmetry, is half the segment LR.

If we omit the isosce­less­ness of blue trian­gles or the pres­ence of right angle in it, the rows in the values triangle won’t be evenly spaced verti­cally. However, in case of self-made model, aban­don­ment of these criteria simpli­fies the construc­tion of monkey’s legs, while nonlin­earity will hardly be notice­able.

The math­e­mat­ical theory of planar hinge mech­a­nisms — lines of different length, hinged at the ends — is still develops today. In 2005, for example, “The Signa­tures Theorem” was proved: for any signa­ture there is a planar hinge mech­a­nism that can precisely repro­duce that signa­ture. But the theorem is not yet construc­tive — it is proved that for every signa­ture such a mech­a­nism exists, but math­e­mati­cians do not yet know how to build one for a partic­ular signa­ture.