- 1303: Chu Shih-Chieh's (1270-1330, China)
transformation method for solving equations (used up to degree 14).
- Meaning, 14th cent. [<>]:
«Supposition theory is developed extensively in its second phase
by logicians such as William of Ockham, Jean Buridan, Gregory of Rimini, and Albert of Saxony.»
- about 1340: William of Ockham formulates Ockham's razor,
the epistemologic principle
that entities should not be multiplied beyond necessity
when trying to explain things [x].
- Logic, 14th cent. [<>]:
«Jean Buridan elaborates a full theory of consequences,
a cross between entailments and inference rules. ...» [x].
- Al-Kashi (1390-1450, Persia)
uses fixed-point iteration to solve a cubic equation.
Glyphs in the 15th century[<>]:
1st half 15th cent.: Italian humanists form antiqua out of
Roman quadratis as capitals
and Carolian minuscles as small letters.
1455: Gutenberg's 45-line bible brings Gothic handwriting to print.
Ca. 1470: Schwabacher script appeared
(another kind of fracture, named after a town close to Nürnberg, Germany).
Used for Luther's bible.
Ca. 1500: Fracture script proper was designed in Augsburg (Germany)
on order of Emperor Maximilian I.
Notat. in the 15th century:
Mid 15th century in France and Italy:
"p" and "m" (with or without a tilde on them) for plus and minus.
In Germany "+" evolved from "&" for "et" (Latin and)
and "-" from the tilde on "mio" (minus) [e.g. Johannes Widman 1489 [TEP]];
used in whole Europe since the late 16th century [HN].
Johann Müller Regiomontanus (German mathematician and astronomer, 1436-76)
introduces the multiplication dot "."
In 1484, Nicolas Chuquet (1445-88, France) uses (free standing!) elevated numbers
for the different powers of a unique variable in formulas [HN].
He is also the first to use (unpublished) negative numbers as coefficients, exponents and solutions.
In 1492, Pellos introduces the decimal point [HN].
Notat. in the 16th century:
In 1557, Robert Recorde (Britisch medician, 1510-58)
invents the equivalence symbol "=" [TEP].
In 1585, Simon Stevin (Dutch merchant and engineer, 1548-1620)
makes decimals (decimal fraction [Math])
popular in Europe, and promotes decimal decimal coinage and measures.
He introduces a notion for real numbers (= scientific notation with E ?),
taken up by Napier[>].
And Thomas Harriot (1560-1621) invented "<" and ">",
recognised negative roots and complex roots.
In 1591[?], François Viète or Vieta (France, 1540-1603, advisor of Henry IV of Navarre)
is one of the first to use systematically (single) letters for known and unknown quantities
(vowels for unknowns, consonants for known quantities;
Descartes uses letters near the beginning of the alphabet for known
and near the end for unknown quantities)
and uses the fraction bar for all divisions [HN].
In 1600, Adriaan van Roomen (1561-1615, Belgium, Germany) uses variable names
with an elevated number for their exponentiation [HN].
(Also used by James Hume 1636 [TEP]).
Encod. + Crypto, 1605:
A 5-bit (``five-level'') code [>]
«used for cryptography by Sir Francis Bacon as far back as 1605»
[Article about revised ASCII]
Non-algorithmic aides to arithmetics begin to appear:
- Aide: [continued from]
1624: "Gunter's scale" to multiply based on logarithms.
1630: Richard Delamain publishes his circular slide rule.
1632: Oughtred's circular slide rule based on two Gunter rulers aids addition, multiplication etc.
Analog, accuracy about 3 digits.
A 1654 slide rule.
Slide rules were used up into the 20th century.
How to use.
- Notat., 1631:
Oughtred [prev. episode]
introduces the multiplication cross "×" [TEP]
(John Wallis 1656(?) uses "×" between numbers, and no symbol between variables)
- Notat., 1639: Stampioen de Jonghe (1610-90, Netherlands)
writes "A:B = C:D" for proportions; since then "=" prevails on the continent
(Did he introduce ":" for proportions?)
- Calculator, 1642-45: Blaise Pascal's (France, 1623-62)
to help tax collecting work adds and subtracts with automatic carry-over
(ignorant of Shickard;
French patent request (1645) and grant (1649)).
- Notat., 1659: Johann Heinrich Rahn (Swiss) introduces ``Pell's division symbol'' "÷" [TEP]
(prevails in the English world since Pell's translation 1668).
- Linguistics, 1660[<>]:
Revival of speculative grammar.
The teachers of Port Royal (France) publish their grammar to prove
that the structure of language is a product of reason.
(1637: Richelieu founded the Académie Française
to regulate vocabulary and grammar of French [EML].
- Notat., 1668 [<>]:
Dalgarno's Characteristica Universalis or Generalis [HSem]:
E.g. 'tan' = conscientia, 'tam' = ingenium, 'taf' = curiositas
where 't' is for the genus of intellectual accients,
and 'a' for the species of actus intellectus primi.
And from 'ska' = religion: 'skaf' = worship and 'skan' = good luck.
But no complete semantic decomposition.
- Notat., 1668 [<>]:
John Wilkins' (Dean of Ripon, later Bishop of Chester)
Real Characters on demand of the Royal Society as a universal script for all languages:
Had a great impact on decyphering and understanding of ancient scripts.
[Maurice Pope: The Story of Decipherment; Thames and Hudson, 1975]
- defines a systematic way to construct symbols for noun(root)s (similar to facette-based archiving).
40 superconcepts (e.g. stone) by first two letters
@ up to 9 subconcepts (e.g. gem) by a consonant
@ 9 species (e.g. turkis) by a vowel.
- + derivational / flexion particles and phonetic symbols.
- Adjectives and verbs are constructed by modification from nouns.
Gottfried Leibniz (1646-1716, Germany).
Calculator, 1673: presents Leibniz's Wheel [Pascalene/Hann's] to the Royal Society (London).
It is the first four function calculator (add, sub, mult, div):
multiplication by automatic repeated addition in accumulator.
Technology: 1694: stepped drum mechanical technology
[next: pin wheel]
in (another?) machine by Leibniz.
Notat.[<] + Principle[<>]:
1666-1679 (published 1701) he developed and strongly advocated positional binary numbers [<]
and binary arithmetics
influenced by a (misunderstood) Llull.
Characteristica Universalis or Generalis, or Lingua Generalis
- his universal ideographic script.
He set out to invent a perfect writing system which would reflect systems of thought directly
and thereby be readable by all human beings regardless of their mother tongues.
1675 he introduces the "
f(x)" notation for his integrals (in print 1686). (Isaac Newton wrote about his integrals introducing the dot notation in 1671, published 1736.
Babbage[>] promotes Leibniz' notation in Britain).
Leibniz persued the symbolism idea that
that (correct) thoughts can be reduced to calculations.
For his Ars Combinatoria,
he encoded concept hierarchies into numbers to enable reasoning by calculation.
«Leibniz used prime numbers to represent conceptual primitives
and multiplied them together to make composite concepts.
... if one concept's number is divisible by that of a second concept.
then the first concept is subsumed by the second (i.e. it is more specific)»
[Ellis, Lehmann: Exploiting the induced order on type-labeled graphs for fast knowledge retrieval; in: Conceptual Structures: Current Practices; LNAI 835, 1994]
Quote from a letter of the old Leibniz [TEP]:
«Ich möchte eine Methode finden ... mit deren Hilfe sich alle Wahrheiten
des Verstandes auf eine Art Rechnung zurückführen lassen.
Diese könnte gleichzeitig eine Art Sprache oder allgemene Ausdrucksform sein,
die sich jedoch von allen bis heute vorgeschlagenen unerscheidet,
da die Zeichen und selbst die Wörter den Verstand leiten
und es sich bei Fehlern (außer den Fehlern der Wirklichkeit)
nur um Rechenfehler handeln würde.
Zwar wäre es sehr schwierig, diese Sprache oder diese Merkmale zu erstellen
oder zu erfinden, doch man könnte sie sehr einfach ohne Wörterbücher erlernen.»
1678/79: He uses the proportion symbol ":"
generally for divisions (prevails in continental European).
1698: Leibniz favors multiplication dot over cross [HN].
1775-1783: American Revolution [Encarta].
Colonial population at that time: approx. 2.5 million people.
Black slaves: more than 500,000 (roughly 22%),
Scots-Irish: ca. 250,000,
Germans: ca. 200,000.
- Model, 1689[<>]:
John Locke (England, 1632-1704) is the person «normally credited with introducing»
the concept of abstraction by which Universals are made (cf. Aristotle).
Locke: «[D]oes it not require some pains and skill to form the general idea of a triangle, ...
for it must be neither Oblique, nor Rectangle, neither Equilateral, Equicural, nor Scalenon;
but all and none of these at once.
In effect it is something imperfect, that cannot exist;
and Idea wherein some parts of several different and inconsistent Ideas are put together».
Therefore abstraction is a feature of human thought [MRW].
- Notat., 17th century Germany: The decimal comma[!]
prevails at school and in buisness (also used by Lagrange 1808)
- Notat., 1706:
Greek letter "" used for the diameter:circumfence proportion
by William Jones [TEP].
- Writer, 1714: Henry Mill's patent on a writing machine (typewriter);
no keyboard, never built [next: typographer]
- Processor, around 1725:
a paper strip was used to control mechanical processes in weaving
mills. In that strip of paper, punch holes were positioned according a certain system. By a
scan mechanism, these punch holes were transferred into mechanical movements
- Notat., about 1862[>]:
Leonhard Euler (1707-83) uses set diagrams to visualize logical inclusions.
Extended by Venn 1880 and then called Venn diagrams.
- Classific., 1749(35/37/53)[<>]:
Carl Linné's (Swedish botanist, 1707-1778)
binomial nomenclature for plants and animals: generic name + specific name [xx].
He grouped genera to classes, and classes to orders [x].
- Processor, 1752:
J F Unger's and J Hohlfeld's (Germany)
processor of piano rolls (Père Engramelle's keyboard transcriptions on paper)
(= punched paper?)
drives a harpsichord [Mus]
- Technology, 17xx-1761:
For determining longitudes at sea,
the British Longitude Act rewards the first clock
that deviates less than 0.2 sec. per day on a ship at sea.
John Harrison (-1776) develops clocks H1, H2,
H3 (1756?, demonstration canceled because of war),
H4 (1761, deviates 5 sec. in 81 days, 8750 pounds reward).
- Calculator, 1770: Philip Mathäus Hahn (priest near Stuttgart)
presents his Staffelwalzen (stepped drum) calculator (``for the spreading of the evangelii'')
An original calculator on display e.g. in the Artihmeum, Bonn (Germany).
[Zahlenrausch; bild der wissenschaft 11/1999; DVA 1999]
== (?) the improved machine of ``Phillip Hann'' in 1774 [hpmuseum] (?)