Mathematicians

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ABC.

In Euclidean geometry any three non-collinear points determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space).

the secondary parts of the triangle

median - a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side

angle bisector - a segment which bisects an angle and whose endpoints are a vertex of the triangle and a point on the opposite side

altitude - a segment from the vertex of the triangle perpendicular to the line containing the opposite side

perpendicular bisector - a line whose points are equidistant from the endpoints of the given side

incenter - the point of concurrency of the three angle bisectors of the triangle

centroid - the point of concurrency of the three medians of the triangle

orthocenter - the point of concurrency of the three altitudes of the triangle

circumcenter - the point of concurrency of the three perpendicular bisectors of the sides of the triangle

You are thinking of Charles Babbage, but no he was not lazy.

His primary concern was error in the calculations and in typesetting the final tables. It took enormous amounts of repetitive and largely redundant work by many human "computers" to calculate and cross check each other's results to find errors and when found to redo the calculation to try to get the right answers, then more work to verify the galley prints from the printshop had not changed the numbers and getting the printshop to retypeset the parts in error. Even after all this work errors remained in the printed tables.

These tables were used for everything at the time (e.g. ship navigation, insurance actuarial calculations, taxes, bank interest, mathematics) and in many of these fields a small error in just one entry of a table could result in significant financial losses!

Charles Babbage designed his Difference Engine (not really a computer) and his later Analytical Engine (a general purpose programmable digital computer) so that they could not make an error without instantly stopping, at which point the operator could either correct the error and restart or if necessary fix the machine's problem causing the error and restart the machine from the beginning. Both machines even typeset the printed tables automatically, so the printshop could not introduce errors after the calculation was done.

However he was never able to get funding to complete either machine.

It took another century and electronics instead of mechanics to give us computers.

The first reported and substantiated use of an abacus, or abacus-like instrument was in Sumeria between 2,700 and 2,300 BCE - or roughly 4,700 years ago.

This used a table of columns with each column equivalent to an order of magnitude above the previous column; just as we would have columns for x10, x100, x1000, x10000.

During the next 2000 years various forms of this came into common usage across areas of the world (Mesopotamia, Persia, Egypt, India, China, Greece, Rome). Some were similar in appearance, while others used the same principals, but looked slightly different. For example the Roman abacus used a clay tablet with columns and counting stones (a counting board), while the Chinese version looked almost identical, but had fixed string columns with counting stones attached to them - similar how most people picture an abacus.

The spread and variance in design of the abacus is probably due to the movement of early traders across the early trade routes (Silk Road etc.). The abacus is still made extensive use of today, as they are simple to use, fast and don't require batteries to operate them. The word abacus comes to us by way of Latin as a mutation of the Greek word abax. In turn, the Greeks may have adopted the Phoenician word abak, meaning "sand," although some authorities lean toward the Hebrew word abhaq, meaning "dust."

Irrespective of the source, the original concept referred to a flat stone covered with sand (or dust) into which numeric symbols were drawn. The first abacus was almost certainly based on such a stone, with pebbles being placed on lines drawn in the sand. Over time, the stone was replaced by a wooden frame supporting thin sticks, braided hair, or leather thongs, onto which clay beads or pebbles with holes were threaded. A variety of different types of abacus were developed, but the most popular became those based on the bi-quinary system, which utilizes a combination of two bases (base-2 and base-5) to represent decimal numbers.

Although the abacus does not qualify as a mechanical calculator, it certainly stands proud as one of first mechanical aids to calculation.

Both the abacus and the counting board are mechanical aids used for counting; they are not calculators in the sense we use the word today. The person operating the abacus performs calculations in their head and uses the abacus as a physical aid to keep track of the sums, the carrys, etc.

What did the first counting board look like?

The earliest counting boards are forever lost because of the perishable materials used in their construction. However, educated guesses can be made about their construction, based on early writings of Plutarch (a priest at the Oracle at Delphi) and others.

In outdoor markets of those times, the simplest counting board involved drawing lines in the sand with ones fingers or with a stylus, and placing pebbles between those lines as place-holders representing numbers (the spaces between 2 lines would represent the units 10s, 100s, etc.). The more affluent people, could afford small wooden tables having raised borders that were filled with sand (usually coloured blue or green). A benefit of these counting boards on tables, was that they could be moved without disturbing the calculation- the table could be picked up and carried indoors.

With the need for portable devices, wooden boards with grooves carved into the surface were then created and wooden markers (small discs) were used as place-holders. The wooden boards then gave way to even more more durable materials like marble and metal (bronze) used with stone or metal markers.
There is no way that anyone can tell who invented the abacus. But it must have been first used as an intermediate way of noting the computation or count before commiting the final result on papyrus for the Egyptians, or on paper or whatever the Greeks used to write their records on.

Remember that the four fundamental operations would have been impossible on both Egypt and Greece's system of writing numbers, but notice that the system of numeration of both are forerunners of the Hindu base 10 system of numeration.

This means that the systems of writing numbers - Hindu, Greek, Egyptian, are in a sense the same. All three write numbers in the 1 to 9, 10 to 90, 100 to 900 patterns.

The numeration system of Greece and Egypt were the cumbersome to use that Rome decided to simplify the writing of numbers, limiting to IVXLCDM and the dash the symbols - overly simplyfying it but emphasizing all the more their need for the Roman Abacus to make their computations.

It could have been the Roman Abacus that served as inspiration to the Chinese Abacus, which is strictly speaking an Hexadecimal Abacus. The Polos, the uncle and father of Marco Polo, who reached China in 1272, must have introduced this innovation to the court of Kublai Khan. One account of the Chinese Abacus mentioned that it first came to notice in the 14th century which is 28 years from 1272. If the Suan Pan became widely used in the mid or late 14th century, that was just enough time for an innovation to filter below from the top, if we are to remember that the Hindu numeration the Arabs brought to Europe via Spain and Italy took several hundred years, from the time Leonardo of Pisa first mentioned it in his book in 1202.

The Filipino Abacus referred to in an earlier note and twisted to sound as if a Filipino invented the abacus, referred to a nine-beaded color-coded by period decimal Filipino Abacus.

Gottfried Leibniz