Mar 17

What is DesCartesian Division?

"If they were alive, how might René Descartes and Isaac Newton explain division to modern mathematics education professors?"

Addition and subtraction are inverse operations. So, it is no surprise elementary mathematics teachers often explain multiplication via repeated addition and division via repeated subtraction. Yet such pedagogies are problematic as they cannot be extended to the set of Real numbers. By contrast, the original ideas and writings of René Descartes and Isaac Newton on multiplication and division easily extend to the Reals. So, if they were alive today, I reveal how they might depict division more deeply and simply than modern mathematics educations professors, in ways currently absent from western K–8 curriculums.

Have fun with the division applet below. Then scroll down to read more about what has gone astray in the development of elementary mathematics pedagogies.


As you can see above, the ‘Cartesian Plane’, named after Descartes, consists of a horizontal x axis and a vertical y axis, that intersect at zero. The plane thus has four quadrants, the first of which is often used for an area model of multiplication. For example, a rectangle drawn with a base of 8 and a height of 3 will cover 24 ‘square units’ on the Cartesian Plane. Thus, 8 multiplied by 3 equals 24, as does 3 multiplied by 8, drawn with a rectangle with a base of 3 and height of 8.

The ‘Cartesian Product’, also named after Descartes, consists of a product set formed from two or more other sets. For example, a child has a set of 8 shirts, each a different colour and a set of 3 skirts, each a different colour. Altogether there are 24 different colour combinations of ‘shirt and skirt’ that can be worn.

Modern mathematics is said to have begun with two great advances from the 1600s. The first was analytic geometry, attributed to Descartes, while the second was calculus, attributed (in priority not publication) to Isaac Newton.

However, neither the Cartesian Plane nor the Cartesian Product has anything to do with the original writings of René Descartes on multiplication. Notably, Newton read Descartes’ 1637 La Géométrie and 1644 Principia Philosophiae (Principles of Philosophy). After this, Newton developed calculus and formulated the laws of motion and universal gravitation published in his 1687 Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). ‘Natural philosophy’ evolved into ‘physical sciences’ and ‘physics’.

Having climbed such scientific heights seldom seen before, Newton went on to write Arithmetica Universalis (Universal Arithmetic) published in 1707. So, having built a reputation as a mathematical and scientific genius, perhaps equalled only by Albert Einstein since in impact, did Newton draw upon repeated addition, equal groups, arrays, or area models to explain multiplication? Or, for that matter, did Descartes? No.

Today, we think nothing of a two-dimensional area being produced by the multiplication of two one-dimensional lines. This is evident from the widespread use of the area model of multiplication and ‘length × width’ calculations. Similarly, a two-dimensional area multiplied by a perpendicular line gives us ‘length × width × height’ which converts the two-dimensional area into a three-dimensional volume.

Since the (c. 820 CE) writings of al-Khwārizmī on algorithms and algebra (derived from his name and book title), the drawing of and solution for equations had been limited to lines, squares and cubes, now associated with the notation x, x2 and x3. There was little progress beyond quadratic (x2) and cubic (x3) equations to quartic (x4) equations for the simple reason a fourth dimension could not be drawn!

The ‘genius idea’ of Descartes, adopted and improved upon by Newton, that formed the foundation of modern mathematics, was the ability to retain consistency of form. Two lines multiplied together could produce, not an area, but a line! Remarkably, this breakthrough, one of the most important in the history of science, appears absent from the minds of modern mathematics education professors. If Descartes and Newton were alive today, they might well argue the aftermath of failing to follow their lead, has been a total failure of educators to understand multiplication and division, let alone teach it.

If Descartes and Newton were alive today, they’d be shocked to witness elementary school teachers being led to believe multiplication IS repeated addition. From the grounding metaphor, or basic mathematical idea ‘Multiplication IS Repeated Addition’ (MIRA), teachers are also led to believe ‘Division IS Repeated Subtraction’ (DIRS). Then, because of this historical and intellectual fraud, students are clumsily led through a multiplicative conceptual minefield of fixes and workarounds to address a simple fact. For Descartes, Newton and even Euclid (fl. 300 BCE), multiplication never was repeated addition and division never was repeated subtraction!


Later this year I will be conducting a workshop in Hungary that calls for a total rethink on how western educators introduce multiplication and division, both more simply than today, and more importantly, in a manner that retains logical and rigorous consistency from the Naturals to the Reals.

Until then, you might like to read or download the following.

A New Model of Multiplication via Euclid

The Lost Logic of Elementary Mathematics and the Haberdasher who Kidnapped Kaizen

Thanks for reading!

Jonathan Crabtree

P.S. Historical turning points relating to co-ordinate geometry include:

  1. Leonardo Pisano assigns the number 1 to a line segment 1220 (Practical Geometry). Previously and for centuries after, people said 1 was not a number, merely a Unit, following Euclid.
  2. Van Ceulen picks up Pisano's ideas circa late 16th C. http://bit.ly/van-ceulen
  3. Descartes pinches Van Ceulen's ideas and blends them with Euclid's. (Naughty, yet clever René!)
  4. Descartes draws similar triangles for multiplication and division and curves in the 1st quarter of our modern cartesian plane in his 1637 La Géométrie.
  5. Van Schooten 'bends' Descartes' acute axes for multiplication/division and makes them right angles.
  6. Newton reads Van Schooten's edition of Descartes' La Géométrie in Latin and by 1680s draws curves in all four quadrants with our modern understanding of positive and negative ordinates/abscissas. These get published in 1704 as an appendix to Opticks. Newton, (like the Indians) accept lines in opposite directions to positive are to be called negative.
  7. Talbot provides commentary on Newton and labels axes with X and Y for the first time in 1860.

P.P.S. There is nothing to stop someone like me reverse engineering mathematics pedagogies by studying the history of mathematics via primary sources. David Hilbert's 1899 Grundlagen der Geometrie (Foundations of Geometry), ALMOST gave the West the above idea of what I call DesCartesian Multiplication. He drew lines as I have done, with a fixed unit segment to create the similar triangles. Yet the mistake Hilbert made was to assume people would extend his idea to the negative reals. now this concept of multiplication on the Reals is complete. Hilbert did not present division via similar triangles in Grundlagen der Geometrie.