... gave measurements of plane figures which agree with the formulas used by Heron, notably those for the equilateral triangle, the regular hexagon (in this case not only the formula but the actual figures agree with Heron's) and the segment of a circle which is less than a semicircle ...

The mechanicians of Heron's school say that mechanics can be divided into a theoretical and a manual part; the theoretical part is composed of geometry, arithmetic, astronomy and physics, the manual of work in metals, architecture, carpentering and painting and anything involving skill with the hands.

... the ancients also describe as mechanicians the wonder-workers, of whom some work by means of pneumatics, as Heron in his Pneumatica, some by using strings and ropes, thinking to imitate the movements of living things, as Heron in his Automata and Balancings, ... or by using water to tell the time, as Heron in his Hydria, which appears to have affinities with the science of sundials.

1. On the dioptra dealing with theodolites and surveying. It contains a chapter on astronomy giving a method to find the distance between Alexandria and Rome using the difference between local times at which an eclipse of the moon is observed at each cities. The fact that Ptolemy does not appear to have known of this method led historians to mistakenly believe Heron lived after Ptolemy;
   
   
2. The pneumatica in two books studying mechanical devices worked by air, steam or water pressure. It is described in more detail below;
   
   
3. The automaton theatre describing a puppet theatre worked by strings, drums and weights;
   
   
4. Belopoeica describing how to construct engines of war. It has some similarities with work by Philon and also work by Vitruvius who was a Roman architect and engineer who lived in the 1st century BC;
   
   
5. The cheirobalistra about catapults is thought to be part of a dictionary of catapults but was almost certainly not written by Heron;
   
   
6. Mechanica in three books written for architects and described in more detail below;
   
   
7. Metrica which gives methods of measurement. We give more details below;
   
   
8. Definitiones contains 133 definitions of geometrical terms beginning with points, lines etc. In [15] Knorr argues convincingly that this work is in fact due to Diophantus;
   
   
9. Geometria seems to be a different version of the first chapter of the Metrica based entirely on examples. Although based on Heron's work it is not thought to be written by him;
   
   
10. Stereometrica measures three-dimensional objects and is at least in part based on the second chapter of the Metrica again based on examples. Again it is though to be based on Heron's work but greatly changed by many later editors;
    
    
11. Mensurae measures a whole variety of different objects and is connected with parts of Stereometrica and Metrica although it must be mainly the work of a later author;
    
    
12. Catoptrica deals with mirrors and is attributed by some historians to Ptolemy although most now seem to believe that this is a genuine work of Heron. In this work, Heron states that vision results from light rays emitted by the eyes. He believes that these rays travel with infinite velocity.

Since 720 has not its side rational, we can obtain its side within a very small difference as follows. Since the next succeeding square number is 729, which has 27 for its side, divide 720 by 27. This gives $26\large\frac{2}{3}\normalsize$. Add 27 to this, making $53\large\frac{2}{3}\normalsize$, and take half this or $26\large\frac{5}{6}\normalsize$. The side of 720 will therefore be very nearly $26\large\frac{5}{6}\normalsize$. In fact, if we multiply $26\large\frac{5}{6}\normalsize$ by itself, the product is $720\large\frac{1}{36}\normalsize$, so the difference in the square is $\large\frac{1}{36}\normalsize$. If we desire to make the difference smaller still than $\large\frac{1}{36}\normalsize$, we shall take $720\large\frac{1}{36}\normalsize$ instead of 729 (or rather we should take $26\large\frac{5}{6}\normalsize$ instead of 27), and by proceeding in the same way we shall find the resulting difference much less than $\large\frac{1}{36}\normalsize$.

if A is the area of a triangle with sides a, b and c and $s = \large\frac{1}{2}\normalsize (a + b + c)$ then
     $A^{2} = s (s - a)(s - b)(s - c)$.

After the measurement of surfaces, rectilinear or not, it is proper to proceed to solid bodies, the surfaces of which we have already measured in the preceding book, surfaces plane and spherical, conical and cylindrical, and irregular surfaces as well. The methods of dealing with these solids are, in view of their surprising character, referred to Archimedes by certain writers who give the traditional account of their origin. But whether they belong to Archimedes or another, it is necessary to give a sketch of these results as well.

Trick jars that give out wine or water separately or in constant proportions, singing birds and sounding trumpets, puppets that move when a fire is lit on an altar, animals that drink when they are offered water ...

The aeolipile was a hollow sphere mounted so that it could turn on a pair of hollow tubes that provided steam to the sphere from a cauldron. The steam escaped from the sphere from one or more bent tubes projecting from its equator, causing the sphere to revolve. The aeolipile is the first known device to transform steam into rotary motion.

The decipherment of the mathematical cuneiform texts made it clear that much of the "Heronic" type of Greek mathematics is simply the last phase of the Babylonian mathematical tradition which extends over 1800 years.

... to me the free flowing, rather discursive style suggests a man well versed in his subject who is giving a quick summary to an audience that knows, or who might be expected to know, a good deal about it.

In the light of recent scholarship, he now appears as a well-educated and often ingenious applied mathematician, as well as a vital link in a continuous tradition of practical mathematics from the Babylonians, through the Arabs, to Renaissance Europe.

The practical utility of Heron's manuals being so great, it was natural that they should have great vogue, and equally natural that the most popular of them at any rate should be re-edited, altered and added to by later writers; this was inevitable with books which, like the "Elements" of Euclid, were in regular use in Greek, Byzantine, Roman, and Arabian education for centuries.