Trident of Newton

Calculus Handbook

Feel like "cheating" at Calculus? Check out our Practically Cheating Calculus Handbook, which gives you hundreds of easy-to-follow answers in a convenient e-book.

Calculus Curves >

trident of newton

The Trident of Newton.

The Trident of Newton (or just “Trident”) is a cubic curve defined by the equation [1]

ax3 + bx2 + cx + d = xy,
where coefficients a, b, c, d, are real constants and a ≠ 0, ad ≠ 0.

The parametric equations and polar equations are, according to Dennis Lawrence [2], “not interesting.” The curve has one discontinuity at x = 0 and one asymptote at x = 0.

The curve, which is sometimes called a Cartesian parabola [3] or Parabola of Descartes (even though it isn’t a parabola) is an esoteric algebraic curve (“algebraic” means that it can be described with polynomials).

The above graph was created with Desmos. Click on the link and you’ll be able to specify different values for the constants.

History of The Trident of Newton

According to MacTutor [4], both Newton and Descartes studied the curve, although it was Newton who named it “trident.” The trident appears in Newton’s Introduction to the Quadrature of Curves. The text was translated into English by by John Harris in 1710 [5]. Newton states that the properties of the trident include:

  • Four infinite legs (i.e., the four ends of the curve tend to infinity),
  • Two tending towards contrary parts (i.e., the y-axis is an asymptote).

Other Meaning of Trident

newton trident

Newton’s trident symbol for bismuth (left) and a sideways trident symbol.

As well as describing a plane curve, Newton used a trident symbol to describe Bismuth. Newton made creative and unorthodox use of alchemical symbols, most of which he created himself [6].


[1] Trident of Newton. Retrieved March 3, 2022 from:
[2] Lawrence, D. (1972). A Catalog of Special Plane Curves. Dover Publications.
[3] Shikin, E. (2014). Handbook and Atlas of Curves. CRC Press.
[4] MacTutor. Trident of Newton. Retrieved March 3, 2022 from:,two%20tending%20towards%20contrary%20parts%20.
[6] Symbol Guide. Retrieved March 3, 2022 from:

Stephanie Glen. "Trident of Newton" From Calculus for the rest of us!

Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!

Leave a Reply

Your email address will not be published.