Mechanical Binary Calculator

A bit of history

The concept of numbers and the counting process developed long before the first historical records, and the way it happened is largely conjectural. However, it’s not difficult to imagine how it probably occurred. It is reasonable to assume that the human species, even in the most primitive times, had some sense of numbers, at least to the point of recognizing more or less when a few objects were added or removed from a small collection. With the gradual evolution of society, simple counting became inevitable.

The oldest method of counting likely involved some form of simple record-keeping, employing the principle of one-to-one correspondence. For counting sheep, for example, one could bend a finger for each animal. People could also count by making notches in clay or on a stone, carving marks on a piece of wood, or tying knots in a rope.

From this perspective, hands become the oldest and most widespread counting accessory. Our first counting machine was digital from the beginning. However, this “computer” was quite limited, even for ancient civilizations that, in certain circumstances, needed many people to count high values.

Our mechanical calculator is binary because it operates with numbers using the binary numeral system. It is precisely this system, with only two distinct symbols for numerical representation (0 and 1), that allows the construction of this Mechanical calculator.

What it is and how it works

The idea of the mechanical calculator is indeed to function as a calculator. It adds numbers represented in the binary system using marbles. 

The operation is fairly simple: 

The marbles represent the digits 0 (absence of a marble) and 1 (presence of a marble). Initially, all flip-flops should be resting on their left side. We place the marbles corresponding to the digit 1 in the representation of the first number and pull the upper mobile plate. We repeat the procedure for the second number. After adjustments (or after the marbles in the machine have settled), we pull the lower mobile plate and obtain the result.

The operation can be seen at: Operação da Calculadora

Now let’s understand the operating principle of this fun calculator!

Firstly, we need to understand a bit about numbers and operations in the binary system or base-2 system. Our usual numbering system is decimal (base 10), meaning we have 10 symbols (the digits from 0 to 9), and numbers are represented in the form:

x_010^0+x_110^1+\cdots+x_n10^n=x_0+x_110+x_2100+\cdots

where x_0, x_1,\ldots , x_n are digits between 0 and 9. Also, note that the digits from 0 to 9 represent precisely the possible remainders in integer division by 10, which is our base.

Translating these ideas to base 2, we have that the possible remainders in division by 2 are 0 or 1, so our symbols will be the digits 0 and 1. Since our base is now 2, the representation will be in the form:

x_02^0+x_12^1+x_22^2+\cdots+x_n2^n=x_0+x_12+x_24+\cdots

For example, let’s add the numbers 17 and 29. To transform them into binary form, we repeatedly divide by 2 until we reach a quotient of zero, obtaining:

Image 2: succecive dividing

When we add them (considering base 2), we get 10001+11101=101110, and converting back to decimals, we have:

17+29=0+1\cdot 2+1 \cdot 4+1 \cdot 8+0\cdot 16+1 \cdot 32=46

Mecanically, the work is done by these movable wooden pieces, which function as physical flip-flops.

In electronics, a flip-flop is a digital circuit capable of serving as a one-bit memory. And that’s exactly what the physical piece does. If the digit inserted in that position is 1, it stores, as seen in figure 2, the corresponding ball for that digit. If in the second inserted number the digit in the position is also 1, the piece will move, discarding the stored ball and storing a ball in the next position, given that the number 2 in the binary system is represented by 10.

When we finish the sum by pulling the lower movable plate, at the same time as we release the balls stored in each position, we put a barrier so that they stop on the wood with the marked positions so that we can obtain our result.

An example for better understanding

Making a very simple sum, like 1+1, let’s observe the operation closely: 

First, we place the marble in position 1 and pull the upper movable plate. Our marble is stored there, next to the flip-flop in position 1.

Placing another marble in position 1 and pulling the upper movable plate again, the flip-flop will turn with the weight of the new marble, releasing the one stored in position 1 (since the lower movable base is not pulled, this marble will be discarded) and storing the new marble in position 2.

Finally, we pull the lower movable plate, and the stored marble will fall into position 2, giving us the result. In the simulator images, the answer appears in the round part of the lower movable plate. In practice, you just need to add up the values of the positions where there is a marble on the lower movable plate after completion.

Figura 6: Finalizando a operação

To make a new sum, just return the two movable plates to their initial position.

Learn more

If you want to test the operation, you can use the simulator.

For those interested and with some knowledge of English, you can also check the details of the machine’s construction, as well as its design.