How To Do Basic Math Operations Using Only Ruler And Compass - An illustration showing what appears like a cartesian plane with measurements being made using a compass to mark two line segments on the positive horizontal axis and the positive vertical axis.

Why would anyone wish to do basic math operations using only ruler and compass? Well, this is what happens when you hang around Euclid for too long. If you are confused, let me elaborate.

In today’s advanced society, children learn counting and calculating using numbers as the fundamental mathematical tools. However, back in Euclid’s day (around 300 B.C.), things were different. Geometry was the fundamental calculating tool available to the people.

What this meant was that in order do basic operations, the only aids that were available were the first three postulates from Euclid’s Elements (arguably the most influential mathematical book series of all time):

1. A straight line segment can be drawn by joining any two points.

2. Any straight line segment can be extended indefinitely into a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre.

– Euclid.

Why Do Math Operations Using Only Ruler and Compass?

If we are to put ourselves in Euclid’s shoes, we have to do all the basic math operations using only ruler and compass (straight lines and circles). But remember, back then, the compass did not even hold it’s legs apart using friction; it folded as soon as it was lifted from the sheet of paper.

To add to that, the ruler was not marked, which means that it could not measure. So, we need to work with an arbitrary unit measurement. It’s intimidating, isn’t it? But worry not. We will approach this challenge one step at a time. Let us begin.

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Addition Using Only Ruler and Compass

Let us begin with an arbitrary unit measurement. Using this unit as the base, let us consider two line segments, each of length ‘l’ and length ‘m’ units respectively.

How To Do Basic Math Operations Using Only Ruler And Compass — An illustration showing two line segments of length ‘l’ and ‘m’ units respecively. ‘m’ seems to be larger than ‘l’.
Line segments of length ‘l’ and ‘m’ — Illustration created by the author

Next, we could just draw a horizontal line with an arbitrary origin ‘O’. Using a compass, with the origin ‘O’ as the centre and length ‘l’ as the radius, we could cut an arc on the horizontal line, giving us the intersection point ‘P’.

With this intersection point ‘P’ as the centre, and with length ‘m’ as the radius, we could mark another arc rightward on the horizontal line, giving us the intersection point ‘Q’. The length of the line segment ‘OQ’ then gives the sum of the two lengths/numbers.

How To Do Basic Math Operations Using Only Ruler And Compass — An illustration showing a horizontal line, with an arbitrary origin ‘O’. Using a compass, with the origin ‘O’ as the centre and length ‘l’ as the radius, we cut an arc to get intersection point ‘P’. With ‘P’ as the centre, and with length ‘m’ as the radius, cut another arc rightward to get intersection point ‘Q’. The line segment OQ gives the the sum of l and m. That is: l+m.
Addition using ruler and compass — Illustrtion created by the author (not drawn to scale)

Now that are we done with addition, let us move on to subtraction.

Subtraction Using Only Ruler and Compass

Let us proceed with the same setup as before. This time, we are interested in computing, say, the difference between the lengths ‘m’ and ‘l’ (m-l).

To achieve this, first, we could use the compass to set the origin ‘O’ as the centre again and length ‘m’ as the radius. Then, we cut an arc to intersect the horizontal line rightward at point ‘Q’.

Now, with ‘Q’ as the centre and length ‘l’ as the radius, we could cut an arc leftward, intersecting the horizontal line at point ‘P’.

How To Do Basic Math Operations Using Only Ruler And Compass — An illustration showing a horizontal line, with an arbitrary origin ‘O’. Using a compass, with the origin ‘O’ as the centre and length ‘m’ as the radius, we cut an arc to get intersection point ‘Q’. With ‘Q’ as the centre, and with length ‘l’ as the radius, cut another arc leftward to get intersection point ‘P’. The line segment OP gives the the difference between m and l. That is: (m minus l).
Subtraction using ruler and compass — Illustrtion created by the author (not drawn to scale)

Given this configuration, the length of line segment ‘OP’ gives the difference between ‘m’ and ‘l’ (m-l). Having conquered subtraction, let us move on to multiplication next.

Multiplication Using Only Ruler and Compass

When it comes to multiplication, things get a little bit tricky. The ancient Greeks used the notion of areas to do multiplication operations. So, strictly speaking, we are doing something more advanced here. But, we are still sticking to Euclid’s postulates. So, all is well!

We first construct a vertical line that is perpendicular to the horizontal line and intersects it at the origin ‘O’ (a make-do Cartesian framework). If you wish to know how to draw perpendicular lines using only a ruler and compass, I’ve briefly covered this in the appendix at the end of this essay. For now, just take my word that it is possible.

Next, we mark the unit lengths using our compass on both these lines from the origin. Let the intersection point on the horizontal line be ‘A’ and the intersection point on the vertical line be ‘B’.

How To Do Basic Math Operations Using Only Ruler And Compass — A compass is used to cut two arcs on the horizontal line — one of unit length with intersection point ‘A’, and another of length ‘l’ with intersection point ‘P’. Similarly, the compass is used to cut two arcs on the vertical line — one of unit length with intersection point ‘B’, and another of length ‘m’ with intersection point ‘Q’.
Multiplication setup — Illustration created by the author (not drawn to scale)

Then, we cut an arc with ‘O’ as the centre and ‘l’ as the radius on the horizontal line, which gives us the intersection point ‘P’. We do a similar procedure for the vertical line but with length ‘m’ to give use the intersection point ‘Q’.

How To Do Basic Math Operations Using Only Ruler And Compass — In the above setup, the points ‘B’ and ‘P’ are now connected using line segment ‘BP’.
Multiplication setup — Illustration created by the author (not drawn to scale)

Now, we connect the points ‘B’ and ‘P’ by constructing a line segment ‘BP’. We are essentially creating a relationship between the vertical line and the horizontal line using the unit length here.

Next, we construct a line segment parallel to line segment ‘BP’ that starts from the point ‘Q’ and intersects the horizontal line at point ‘C’. This gives us the line segment ‘QC’.

How To Do Basic Math Operations Using Only Ruler And Compass — In the above setup, a parallel line segment to line segment ‘BP’ is constructed from point Q to the horizontal line. The intersection point on the horitontal line is ‘C’. The line segment ‘OC’, then, gives the product of l and m. That is: (l*m).
Multiplication setup — Illustration created by the author (not drawn to scale)

A parallel line is perpendicular to a perpendicular line. Constructing it with only ruler and compass is a hassle, but possible. The length of the line segment ‘OC’, then, gives us the product of ‘l’ and ‘m’ (l*m).

To understand why this is, first note that the triangles ‘OBP’ and ‘OQC’ are similar triangles. Consequently, the following relationship holds:

OB/OP = OQ/OC

When we plug in the known length values, we get the following equation:

1/l = m/OC

Cross multiplying the terms, we get:

OC = (l*m).

Division Using Only Ruler and Compass

Let us say that we are interested in computing (m/l). To do this, we start with a similar setup as we had with multiplication:

How To Do Basic Math Operations Using Only Ruler And Compass — A compass is used to cut two arcs on the horizontal line — one of unit length with intersection point ‘A’, and another of length ‘l’ with intersection point ‘P’. Similarly, the compass is used to cut two arcs on the vertical line — one of unit length with intersection point ‘B’, and another of length ‘m’ with intersection point ‘Q’.
Division setup — Illustration created by the author (not drawn to scale)

This time, we connect the points ‘Q’ and ‘P’ to create the line segment ‘QP’. We are essentially creating a relationship between the vertical line and the horizontal line using the lengths ‘l’ and ‘m’ this time.

How To Do Basic Math Operations Using Only Ruler And Compass — In the above setup, the points ‘Q’ and ‘P’ are now connected using line segment ‘QP’.
Division setup — Illustration created by the author (not drawn to scale)

Next, we construct a parallel line segment to the line segment ‘QP’ that starts from the point ‘P’ and cuts the vertical line at point ‘C’. The length of the line segment ‘OC’, then, gives us the value of (m/l).

How To Do Basic Math Operations Using Only Ruler And Compass — In the above setup, a parallel line segment to line segment ‘QP’ is constructed from point P to the vertical line. The intersection point on the vertical line is ‘C’. The line segment ‘OC’, then, gives the value of m over l. That is: (m/l).
Division setup — Illustration created by the author (not drawn to scale)

To understand why this is, note that the triangles ‘OCA’ and ‘OQP’ are similar triangles. Consequently, the following relationship holds:

OQ/OP = OC/OA

When we plug in the known lengths, we get the following equation:

m/l = OC/1

Therefore:

OC = m/l

There you go. We just did the basic mathematical operations of addition, subtraction, multiplication, and division using only a ruler and compass.

Final Remarks

One of the reasons why I decided to cover this topic is that it forms the base for a more profound challenge that Euclid faced. Here is a list of questions that Euclid and his disciples were directly or indirectly trying to answer:

1. Can you trisect an angle?

2. Can you double a cube?

3. Is it possible to square a circle?

It would not be possible for me to cover such topics without having covered the basic principles first. Hence, this essay. So, you can expect a more advanced follow-up essay that answers the above questions sometime soon!

Appendix — Constructing a Perpendicular Line

Let us say that we have a line segment ‘AB’ of arbitrary length ‘AB’. Using a compass, we first take the point ‘A’ as the centre and length ‘AB’ as the radius and construct a circle.

How To Do Basic Math Operations Using Only Ruler And Compass — An illustration showing a line segment ‘AB’ with end points ‘A’ and ‘B’. Using a compass, ‘A’ is held as the centre and with ‘AB’ as the radius, a circle is drawn.
Constructing a perpendicular line segment — illustration created by the author (not drawn to scale)

Next, take the point ‘B’ as the centre and length ‘BA’ as the radius and construct another circle. These two circles will have two intersection points ‘C’ and ‘D’.

How To Do Basic Math Operations Using Only Ruler And Compass —In the above setup, now, using the compass, ‘B’ is taken as the centre and with AB as the radius, another circle is drawn. The two circles have two intersection points ‘C’ and ‘D’. The line segment ‘CD’ joining these two points is not only perpendicular to ‘AB’, but also bisects the line segment.
Constructing a perpendicular line segment — illustration created by the author (not drawn to scale)

The line segment ‘CD’ that connects these points is not only perpendicular to the line segment ‘AB’, but also bisects it.

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