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 NCB Deposit  # 103

Janet Beery
University of Redlands

Lou Talman
Metropolitan State Univ.

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The Cannonball Curves of Thomas Harriot 

Projectile Motion circa 1600

Thomas Harriot

  The English mathematician Thomas Harriot is best known for accomplishments in three areas.
* Navigation
* Astronomy
* Algebra
We now investigate his models of projectile motion; specifically, determining the path of a cannon or mortar shot.

Harriot’s first model (above) predicted the paths traced in red for shots fired at elevation angles of 30, 45, and 60 degrees.  Each of these paths has the same maximum height and, as the elevation angle decreases, the range increases.  Harriot’s final model, predicting the curves traced in blue, was much more realistic.  This model matches both reality and the curves predicted by classical mechanics very well.

Harriot's patron, Sir Walter Raleigh, sent Harriot on a voyage to the New World in 1585.  Harriot lived in what is now North Carolina for one year before sailing back to England with Sir Francis Drake in 1586.  He wrote a book about his experiences called "A Briefe and True Report of the New Found Land of Virginia" that was translated into many languages.

Beginning in July of 1609, Harriot made drawings of the moon, as observed through his brand-new telescope.  This was five months before Galileo would do the same and gain much greater recognition for it.

Earlier contributions from Tartaglia's "Nova scientia . . ." (1537):

Tartaglia cannon

Harriot on Projectile Motion:  The Cannon Shot
Harriot's starting point was the medieval belief, presented in the gunnery manuals of his time, that the flight of the cannonball began with "violent motion" out of the muzzle of the gun and ended with "natural motion" as it fell to the ground.  Anyone who had seen a cannon or mortar shot knew that the motion did not abruptly change from one to the other, but rather made a more gradual transition from "violent" to "natural" motion during a period of "mixed" motion.

Harriot would construct cannonball curves by plotting points marking the cannonball's position in the air after equal intervals of time, and then drawing a smooth curve connecting the points.  By taking into account both motion in the direction of the shot and motion straight downwards due to gravity, he would combine "violent" and "natural" motion into smooth curves.


Harriot's First Models
Angle of 30 degrees

Angle of 45 degrees

Angle of 60 degrees

How realistic are Harriot's First Models?

Study the above curves.  The shape for a shot at an elevation angle of 30 to 60 degrees is fairly realistic.  However, you can see already that all trajectories seem to have the same height and that the range of the shot increases as the elevation angle decreases.

Harriot's Final Models
* Motion in the direction of the shot decelerates according to the sequence 15, 13, 11, 9, 7, 5, 3, 1
* Vertical motion accelerates downward according to the sequence 1, 3, 5, 7, 9, 11, 13, 15
Angle of 30 degrees

Angle of 45 degrees

Angle of 60 degrees

How realistic are Harriot's Final Models?

Study the above curves.  These models agree exceptionally well with the results predicted by the classical mechanics introduced by Isaac Newton (1642-1727) and still taught in physics courses today.  However, Harriot did not use equations but rather constructions similar to those given above.

References and Comments

Scholars have some doubts about the authenticity of the image of Thomas Harriot shown in the upper right.

Among the new notation introduced in Harriot's algebra book Artis Analyticae Praxis published by his friends in 1631, ten years after his death, were the symbols for less than and greater than, < and >, that we use today.

Thomas Harriot :

Moon drawings >

Writings on North Carolina
 < >

Artis Analyticae praxis, ad aequations algebraïcas . . . , Apud Robertum Barker, London, 1631.

< >
Schemmel, Matthias, The English Galileo:  Thomas Harriot's Work on Motion as an Example of Preclassical Mechanics (2 vols.), Springer, 2008.
This deposit is an attempt to present Harriot's and Schemmel's work in a nutshell - or perhaps we should say, "in a mortar-shell."
The authors are grateful to the Huntington Library for permission to use the illustrations in this article.  <>
The animations created in Quick Time in 2010
were converted to "mp_4" format in 2018.

Mathematics Talman created individual images for
the Quick Time Movie using
Wolfram Mathematica®.
siglou     2010
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