Ancient Greece was an advanced civilization that used mathematics to produce both unique and devastating military weapons which changed the outcome of war as we know it. From the early ages Ancient Greece created weapons such as boats, towers, and artillery. Although many documents of mathematical progression do not exist, we still are able to acknowledge the advancements made through the use of geometry.

Gastraphetes

The Gastraphetes was an artillery weapon used by the Romans which was created around 397 B.C. by an individual named Biton (Lahanas). This device could fire stone which weighed up-to 40 pounds nearly 300 - 400ft (Lahanas). Biton used Geometry and measurements such as seen in his weapon plans on the next slide.

Cheiroballistra

Another magnificent device which was created for military purposes was the Cheiroballistra. This weapon was created in 100 AD and was one of the first artillery weapons constructed with metal frames capable of hurling arrows at a long distance (Lahanas). This device was constructed by Apollodorus, another inventor and mathematician. Apollodorus just like Biton used Geometry to create this weapon utilizing calculations of distance and angels to perform accurate damage to its enemy.

Introduction:

Since the beginning of military assembly, math has played a key-role in the development of: technology, weapons, and communication within the armed forces. From the early ages of Ancient Greece, to the destructive era of World War II, and the current conflicts which persist today, it is because of the equations mathematicians have formulated that counties have been able to defend themselves from foreign powers utilizing technology.

**History of Math in the Military**

**By: Andrew C. Ducharme**

Helepolis

The Helepolis was an invention built in Ancient Greece in the year 304 B.C. (Lahanas). This was created and engineered by an individual named Polyidus of Thessaly for the use of military purposes (Lahanas). Although this weapon was fairly simple, Polyidus used simple geometry to create the structure. You can see this as there were many points, lines, measurements, and angles in the diagram. These characteristics are then used to lay out the building in accordance to the drawing.

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Artillery

http://www.mlahanas.de/Greeks/war/CatapultTypes.htm

http://www.mlahanas.de/Greeks/war/CatapultTypes.htm

Construction Plans

http://www.storyofmathematics.com/greek.html

http://www.mlahanas.de/Greeks/war/CatapultTypes.htm

Wheeled Tower

http://www.mlahanas.de/Greeks/war/Helepolis.htm

http://www.allposters.com/-sp/Tower-of-Demetrius-Poliorcetes-During-the-Siege-of-Rhodes-in-305-BC-Posters_i1586632_.htm?AID=48653529

Boats

http://www.artsales.com/ARTistory/Ancient_Ships/07_merchant_ships.html

Super Warships

Another remarkable advancement made by the Ancient Greeks were warships also known as a Triremes. These warships were made using their most current mathematics at the time creating a vessel which would require 170 men to row; one on each oar ("Ancient Greeks..."). Another interesting fact about Triremes were that when created, they were measured at 115ft long and were the fastest ships of the time ("Ancient Greeks..."). Just like the Helepolis, Boats were constructed using simple geometry to calculate lines, points, measurements, and angles.

These advancements in Ancient Greece among others prove loyal to the mathematics progression of the time. Although these individuals did not have the advancements as we do today it is because of the improvement of Geometry which Ancient Greece adopted from both Egypt and Babylon that the military's of the future were able to further advance its weaponry.

World War II

http://rising.blackstar.com/photos-sear-the-memory-in-a-way-that-video-does-not.html/ww2-156-l

World War II was one of the most bloodiest wars in recorded history. Although this war went unjust, it is because of the advancements made through technology by mathematical contributions that we were able to successfully diminish our foreign enemies through advancements contributed by mathematics and mathematicians. Advancements include the Atomic Bomb, aircraft's, and computers.

Atomic Bomb

http://www.tiki-toki.com/timeline/entry/287123/The-Most-Controversial-Decision-The-Atomic-Bombs/

The Atomic bomb which was created in 1945 was a weapon created by Albert Einstein, a mathematical genius. With his famous formula known as E=mc2, he was able to create an incredible weapon. This was used for military purposes during World War II when facing against Japan. Without his brilliant work and the creation of the Atomic Bomb, the outcome of World War II could have severely differed.

Aircraft's

http://www.ieeeghn.org/wiki/index.php/File:B29_bomber.JPG

Aircraft's during World War II were an incredible advancement during its time. Throughout the war, aircraft's such as bombers, reconnaissance, and fighters played a key role in its success ("World War II..."). New type of aircraft's were beginning to emerge, with propelled aircraft's now utilizing jet-engines ("World War II..."). Advancements to its inner technology began erupting too, with radar being utilized inside aircraft's, specifically the Spitfire, an advanced fighter made by Britain ("World War II..."). One equation which was specifically made for radar was the radar range equation, an equation which estimates the power at the input to the receiver for a target of a given radar cross section at a specified range ("Documentation"). This was a algebraic formula, which shows one of the current forms of mathematics used at the time.

Original Computers

ENIAC

Electronic Numerical Integrator And Computer, also known as (ENIAC) was originally created in 1946 to calculate artillery firing tables for the U.S. Army (Aue 63). In addition to calculating artillery firing tables, ENIAC was also found to be capable of solving numerical equations. Overall, ENIAC was known as the most advanced calculator of the century ("ENIAC...") This was an incredible advancement in both math and the military. With this technology, the military was able to become more efficient, and also gave new possibilities to the future of warfare.

Enigma

Another excellent device used by the military in World War II was the Encryption System, also known as Enigma (Murray 853). The Enigma system was constructed somewhere around 1920s, and produced by a German engineer named Arthur Scherbius (Lycett). This device was used to encipher a message before being transmitted to its intended target (Lycett). Enigma used permutations to generate it's cipher, which follows under the Group Theory (Tuma 174). Although the enemy held this device too, Polish mathematicians named Marian Rejewski, Henryk Zygalski, and Jerzy Rozycki broke the encryption code (Tuma 14). This was an all around important discovery as it gave allies the upper hand in predicting movement made by enemy forces.

Encryption System

http://www.bbc.co.uk/history/topics/enigma

These advancements made in World War II are due to the mathematical progression overtime. It is because of these brilliant mathematicians that we have improved our military warfare. The Atomic Bomb could not have been discovered without Albert Einsteins famous equation E=mc2, the advancement of aircraft's and radar without Britain, or the advanced encryption system known as Enigma discovered by Arthur Scheribus (Lycett;"World War II..."). These individuals are all contributors the advancement of math in military history.

Modern Day Military

Missiles

http://www.naval-technology.com/projects/naresuan-class-guided-missile-frigates/naresuan-class-guided-missile-frigates4.html

http://www.politico.com/story/2014/02/warships-sochi-olympics-security-103171.html

Drones

http://dronewars.net/aboutdrone/

Conclusion:

Today, drones play a huge roll in the military. These aircraft's which are unmanned, use advanced mathematical tools to perform reconnaissance and armed attacks which are far more precise than that of the human mind. Although much of the information on the mathematical tools used are classified, you can predict that Drones use many mathematical equations that a human being would utilize to calculate distance, elevation, and speed. Furthermore, as mathematics continue to progress and calculations become more precise, changes to drones will continue making them far more reliable, accurate, and advanced.

Missiles, another device used by the military, uses targeting systems and indicators to accurately reach its destination. Missiles use common formulas such as the Pythagorean Theorem which calculates distance from point (A) to point (B). This is a very well known mathematical equation that is important in military mechanics, as it gives an estimated time of impact and whether it will have a sufficient enough of time to hit its intended target.

Modern Day mathematical equations have advanced further than that of seen in Ancient Greece or World War II. It is because of the discoveries made previously that mathematicians are able to further understand and utilize formulas such as seen in Geometry and Algebra.

This is how history of math has played a key-role in the development of military technology throughout the centuries. From the first artillery device seen in Ancient Greece, to the Atomic Bomb in World War II, and the current technology which persists today, it is with the mathematical formulas created that these advancements in the military are able to be made.