History of geometry

Source: Wikipedia, the free encyclopedia.
Cyclopaedia
.

Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers (arithmetic
).

Classic geometry was focused in

axiomatic method still in use today. His book, The Elements is widely considered the most influential textbook of all time, and was known to all educated people in the West until the middle of the 20th century.[1]

In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. (See

.)

Early geometry

The earliest recorded beginnings of geometry can be traced to early peoples, such as the

Sulba Sutras around 800 BC contained the first statements of the theorem; the Egyptians had a correct formula for the volume of a frustum
of a square pyramid.

Egyptian geometry

The ancient Egyptians knew that they could approximate the area of a circle as follows:[2]

Area of Circle ≈ [ (Diameter) x 8/9 ]2.

Problem 50 of the Ahmes papyrus uses these methods to calculate the area of a circle, according to a rule that the area is equal to the square of 8/9 of the circle's diameter. This assumes that π is 4×(8/9)2 (or 3.160493...), with an error of slightly over 0.63 percent. This value was slightly less accurate than the calculations of the Babylonians (25/8 = 3.125, within 0.53 percent), but was not otherwise surpassed until Archimedes' approximation of 211875/67441 = 3.14163, which had an error of just over 1 in 10,000.

Ahmes knew of the modern 22/7 as an approximation for π, and used it to split a hekat, hekat x 22/x x 7/22 = hekat;[citation needed] however, Ahmes continued to use the traditional 256/81 value for π for computing his hekat volume found in a cylinder.

Problem 48 involved using a square with side 9 units. This square was cut into a 3x3 grid. The diagonal of the corner squares were used to make an irregular octagon with an area of 63 units. This gave a second value for π of 3.111...

The two problems together indicate a range of values for π between 3.11 and 3.16.

Problem 14 in the Moscow Mathematical Papyrus gives the only ancient example finding the volume of a frustum of a pyramid, describing the correct formula:

where a and b are the base and top side lengths of the truncated pyramid and h is the height.

Babylonian geometry

The Babylonians may have known the general rules for measuring areas and volumes. They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if π is estimated as 3. The volume of a cylinder was taken as the product of the base and the height, however, the volume of the frustum of a cone or a square pyramid was incorrectly taken as the product of the height and half the sum of the bases. The Pythagorean theorem was also known to the Babylonians. Also, there was a recent discovery in which a tablet used π as 3 and 1/8. The Babylonians are also known for the Babylonian mile, which was a measure of distance equal to about seven miles today. This measurement for distances eventually was converted to a time-mile used for measuring the travel of the Sun, therefore, representing time.[3] There have been recent discoveries showing that ancient Babylonians may have discovered astronomical geometry nearly 1400 years before Europeans did.[4]

Vedic India geometry

Rigveda manuscript in Devanagari.

The Indian Vedic period had a tradition of geometry, mostly expressed in the construction of elaborate altars. Early Indian texts (1st millennium BC) on this topic include the

Satapatha Brahmana and the Śulba Sūtras.[5][6][7]

According to (Hayashi 2005, p. 363), the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians."

The diagonal rope (akṣṇayā-rajju) of an oblong (rectangle) produces both which the flank (pārśvamāni) and the horizontal (tiryaṇmānī) <ropes> produce separately."[8]

They contain lists of

Diophantine equations.[10]
They also contain statements (that with hindsight we know to be approximate) about squaring the circle and "circling the square."[11]

The

Baudhayana Sulba Sutra
, the best-known and oldest of the Sulba Sutras (dated to the 8th or 7th century BC) contains examples of simple Pythagorean triples, such as: , , , , and [12] as well as a statement of the Pythagorean theorem for the sides of a square: "The rope which is stretched across the diagonal of a square produces an area double the size of the original square."[12] It also contains the general statement of the Pythagorean theorem (for the sides of a rectangle): "The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together."[12]

According to mathematician S. G. Dani, the Babylonian cuneiform tablet Plimpton 322 written c. 1850 BC[13] "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple,[14] indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850 BC. "Since these tablets predate the Sulbasutras period by several centuries, taking into account the contextual appearance of some of the triples, it is reasonable to expect that similar understanding would have been there in India."[15] Dani goes on to say:

"As the main objective of the Sulvasutras was to describe the constructions of altars and the geometric principles involved in them, the subject of Pythagorean triples, even if it had been well understood may still not have featured in the Sulvasutras. The occurrence of the triples in the Sulvasutras is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and would not correspond directly to the overall knowledge on the topic at that time. Since, unfortunately, no other contemporaneous sources have been found it may never be possible to settle this issue satisfactorily."[15]

In all, three Sulba Sutras were composed. The remaining two, the Manava Sulba Sutra composed by

Apastamba
(c. 600 BC), contained results similar to the Baudhayana Sulba Sutra.

Greek geometry

Classical Greek geometry

For the ancient Greek mathematicians, geometry was the crown jewel of their sciences, reaching a completeness and perfection of methodology that no other branch of their knowledge had attained. They expanded the range of geometry to many new kinds of figures, curves, surfaces, and solids; they changed its methodology from trial-and-error to logical deduction; they recognized that geometry studies "eternal forms", or abstractions, of which physical objects are only approximations; and they developed the idea of the "axiomatic method", still in use today.

Thales and Pythagoras

Pythagorean theorem: a2 + b2 = c2

Thales (635-543 BC) of Miletus (now in southwestern Turkey), was the first to whom deduction in mathematics is attributed. There are five geometric propositions for which he wrote deductive proofs, though his proofs have not survived. Pythagoras (582-496 BC) of Ionia, and later, Italy, then colonized by Greeks, may have been a student of Thales, and traveled to Babylon and Egypt. The theorem that bears his name may not have been his discovery, but he was probably one of the first to give a deductive proof of it. He gathered a group of students around him to study mathematics, music, and philosophy, and together they discovered most of what high school students learn today in their geometry courses. In addition, they made the profound discovery of incommensurable lengths and irrational numbers
.

Plato

trisect an angle, to construct a cube twice the volume of a given cube, and to construct a square equal in area to a given circle. The proofs of the impossibility of these constructions, finally achieved in the 19th century, led to important principles regarding the deep structure of the real number system. Aristotle (384-322 BC), Plato's greatest pupil, wrote a treatise on methods of reasoning used in deductive proofs (see Logic
) which was not substantially improved upon until the 19th century.

Hellenistic geometry

Euclid

Statue of Euclid in the Oxford University Museum of Natural History.
Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310)

Hellenistic mathematicians knew at the time about geometry; Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid's was not the first elementary geometry textbook, but it was so much superior that the others fell into disuse and were lost. He was brought to the university at Alexandria by Ptolemy I
, King of Egypt.

The Elements began with definitions of terms, fundamental geometric principles (called axioms or postulates), and general quantitative principles (called common notions) from which all the rest of geometry could be logically deduced. Following are his five axioms, somewhat paraphrased to make the English easier to read.

  1. Any two points can be joined by a straight line.
  2. Any finite straight line can be extended in a straight line.
  3. A circle can be drawn with any center and any radius.
  4. All right angles are equal to each other.
  5. If two straight lines in a plane are crossed by another straight line (called the transversal), and the interior angles between the two lines and the transversal lying on one side of the transversal add up to less than two right angles, then on that side of the transversal, the two lines extended will intersect (also called the parallel postulate).

Concepts, that are now understood as

Greek geometric algebra
.

Archimedes

Measurement of the Circle and On Conoids and Spheroids. His work On Floating Bodies is the first known work on hydrostatics, of which Archimedes is recognized as the founder. Renaissance translations of his works, including the ancient commentaries, were enormously influential in the work of some of the best mathematicians of the 17th century, notably René Descartes and Pierre de Fermat.[17]

After Archimedes

compass in this 13th-century manuscript is a symbol of God's act of Creation
.

After Archimedes, Hellenistic mathematics began to decline. There were a few minor stars yet to come, but the golden age of geometry was over. Proclus (410-485), author of Commentary on the First Book of Euclid, was one of the last important players in Hellenistic geometry. He was a competent geometer, but more importantly, he was a superb commentator on the works that preceded him. Much of that work did not survive to modern times, and is known to us only through his commentary. The Roman Republic and Empire that succeeded and absorbed the Greek city-states produced excellent engineers, but no mathematicians of note.

The great Library of Alexandria was later burned. There is a growing consensus among historians that the Library of Alexandria likely suffered from several destructive events, but that the destruction of Alexandria's pagan temples in the late 4th century was probably the most severe and final one. The evidence for that destruction is the most definitive and secure. Caesar's invasion may well have led to the loss of some 40,000-70,000 scrolls in a warehouse adjacent to the port (as Luciano Canfora argues, they were likely copies produced by the Library intended for export), but it is unlikely to have affected the Library or Museum, given that there is ample evidence that both existed later.[18]

Civil wars, decreasing investments in maintenance and acquisition of new scrolls and generally declining interest in non-religious pursuits likely contributed to a reduction in the body of material available in the Library, especially in the 4th century. The Serapeum was certainly destroyed by Theophilus in 391, and the Museum and Library may have fallen victim to the same campaign.

Classical Indian geometry

In the Bakhshali manuscript, there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero."[19] Aryabhata's Aryabhatiya (499) includes the computation of areas and volumes.

Brāhma Sphuṭa Siddhānta in 628. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain).[20] In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral:[20]

Brahmagupta's theorem: If a cyclic quadrilateral has diagonals that are perpendicular to each other, then the perpendicular line drawn from the point of intersection of the diagonals to any side of the quadrilateral always bisects the opposite side.

Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of

rational triangles
(i.e. triangles with rational sides and rational areas).

Brahmagupta's formula: The area, A, of a cyclic quadrilateral with sides of lengths a, b, c, d, respectively, is given by

where s, the semiperimeter, given by:

Brahmagupta's Theorem on rational triangles: A triangle with rational sides and rational area is of the form:

for some rational numbers and .[21]

Lhuilier [1782], 350 years later. With the sides of the cyclic quadrilateral
being a, b, c, and d, the radius R of the circumscribed circle is:

Chinese geometry

Nine Chapters on the Mathematical Art, first compiled in 179 AD, with added commentary in the 3rd century by Liu Hui
.
Haidao Suanjing, Liu Hui, 3rd century.

The first definitive work (or at least oldest existent) on geometry in China was the

Qin Shihuang
(r. 221-210 BC), multitudes of written literature created before his time were purged. In addition, the Mo Jing presents geometrical concepts in mathematics that are perhaps too advanced not to have had a previous geometrical base or mathematic background to work upon.

The Mo Jing described various aspects of many fields associated with physical science, and provided a small wealth of information on mathematics as well. It provided an 'atomic' definition of the geometric point, stating that a line is separated into parts, and the part which has no remaining parts (i.e. cannot be divided into smaller parts) and thus forms the extreme end of a line is a point.

atomists of Democritus, the Mo Jing stated that a point is the smallest unit, and cannot be cut in half, since 'nothing' cannot be halved.[24] It stated that two lines of equal length will always finish at the same place,[24] while providing definitions for the comparison of lengths and for parallels,[25] along with principles of space and bounded space.[26] It also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch.[27] The book provided definitions for circumference, diameter, and radius, along with the definition of volume.[28]

The

Franciscus Vieta
(1540-1603) fell halfway between Zu's approximations.

The Nine Chapters on the Mathematical Art

tetrahedral wedge.[34] He also figured out that a wedge with trapezoid base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid.[34] Furthermore, Liu Hui described Cavalieri's principle on volume, as well as Gaussian elimination
. From the Nine Chapters, it listed the following geometrical formulas that were known by the time of the Former Han Dynasty (202 BCE–9 CE).

Areas for the[35]

Volumes for the[34]

Continuing the geometrical legacy of ancient China, there were many later figures to come, including the famed astronomer and mathematician

Pascal's Triangle, Xu Guangqi
(1562-1633), and many others.

Islamic Golden Age

Al-Jabr wa-al-Muqabilah

By the beginning of the 9th century, the "Islamic Golden Age" flourished, the establishment of the House of Wisdom in Baghdad marking a separate tradition of science in the medieval Islamic world, building not only Hellenistic but also on Indian sources.

Although the Islamic mathematicians are most famed for their work on

number systems, they also made considerable contributions to geometry, trigonometry and mathematical astronomy, and were responsible for the development of algebraic geometry.[citation needed
]

Al-Karaji (born 953) completely freed algebra from geometrical operations and replaced them with the arithmetical type of operations which are at the core of algebra today.[citation needed]

In some respects,

special right triangles to all triangles in general, along with a general proof.[36]

al-Quhi (born 940) were leading figures in a revival and continuation of Greek higher geometry in the Islamic world. These mathematicians, and in particular Ibn al-Haytham, studied optics and investigated the optical properties of mirrors made from conic sections.[citation needed
]

Astronomy, time-keeping and

]

A 2007 paper in the journal Science suggested that

Renaissance

compass
here is an icon of religion as well as science, in reference to God as the architect of creation

The

Arabic literature of the 9th to 10th century "Islamic Golden Age" began in the 10th century and culminated in the Latin translations of the 12th century
. A copy of Ptolemy's Almagest was brought back to Sicily by Henry Aristippus (d. 1162), as a gift from the Emperor to King William I (r. 1154–1166). An anonymous student at Salerno travelled to Sicily and translated the Almagest as well as several works by Euclid from Greek to Latin.[39] Although the Sicilians generally translated directly from the Greek, when Greek texts were not available, they would translate from Arabic. Eugenius of Palermo (d. 1202) translated Ptolemy's Optics into Latin, drawing on his knowledge of all three languages in the task.[40] The rigorous deductive methods of geometry found in Euclid's Elements of Geometry were relearned, and further development of geometry in the styles of both Euclid (Euclidean geometry) and Khayyam (algebraic geometry) continued, resulting in an abundance of new theorems and concepts, many of them very profound and elegant.

Advances in the treatment of perspective were made in Renaissance art of the 14th to 15th century which went beyond what had been achieved in antiquity. In

Basilica di San Lorenzo in Florence by Filippo Brunelleschi (1377–1446).[41]

In c. 1413 Filippo Brunelleschi demonstrated the geometrical method of perspective, used today by artists, by painting the outlines of various Florentine buildings onto a mirror. Soon after, nearly every artist in Florence and in Italy used geometrical perspective in their paintings,

Loreto, Forlì and others), and was celebrated for that. Not only was perspective a way of showing depth, it was also a new method of composing
a painting. Paintings began to show a single, unified scene, rather than a combination of several.

As shown by the quick proliferation of accurate perspective paintings in Florence, Brunelleschi likely understood (with help from his friend the mathematician

Biagio Pelacani da Parma
who studied Alhazen's Optics.

Platonic solids
as they would appear in perspective.

Perspective remained, for a while, the domain of Florence.

The Arnolfini Portrait
, because he was unaware of the theoretical breakthrough just then occurring in Italy. However he achieved very subtle effects by manipulations of scale in his interiors. Gradually, and partly through the movement of academies of the arts, the Italian techniques became part of the training of artists across Europe, and later other parts of the world. The culmination of these Renaissance traditions finds its ultimate synthesis in the research of the architect, geometer, and optician
Girard Desargues on perspective, optics and projective geometry.

The

De Architectura
.

Modern geometry

The 17th century

Discourse on Method by René Descartes

In the early 17th century, there were two important developments in geometry. The first and most important was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics. The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry is the study of geometry without measurement, just the study of how points align with each other. There had been some early work in this area by Hellenistic geometers, notably Pappus (c. 340). The greatest flowering of the field occurred with Jean-Victor Poncelet (1788–1867).

In the late 17th century, calculus was developed independently and almost simultaneously by Isaac Newton (1642–1727) and Gottfried Wilhelm Leibniz (1646–1716). This was the beginning of a new field of mathematics now called analysis. Though not itself a branch of geometry, it is applicable to geometry, and it solved two families of problems that had long been almost intractable: finding tangent lines to odd curves, and finding areas enclosed by those curves. The methods of calculus reduced these problems mostly to straightforward matters of computation.

The 18th and 19th centuries

Non-Euclidean geometry

The very old problem of proving Euclid's Fifth Postulate, the "

Saccheri, Lambert, and Legendre each did excellent work on the problem in the 18th century, but still fell short of success. In the early 19th century, Gauss, Johann Bolyai, and Lobachevsky, each independently, took a different approach. Beginning to suspect that it was impossible to prove the Parallel Postulate, they set out to develop a self-consistent geometry in which that postulate was false. In this they were successful, thus creating the first non-Euclidean geometry. By 1854, Bernhard Riemann, a student of Gauss, had applied methods of calculus in a ground-breaking study of the intrinsic (self-contained) geometry of all smooth surfaces, and thereby found a different non-Euclidean geometry. This work of Riemann later became fundamental for Einstein's theory of relativity
.

is shown as 'divine geometer' (1795)

It remained to be proved mathematically that the non-Euclidean geometry was just as self-consistent as Euclidean geometry, and this was first accomplished by Beltrami in 1868. With this, non-Euclidean geometry was established on an equal mathematical footing with Euclidean geometry.

While it was now known that different geometric theories were mathematically possible, the question remained, "Which one of these theories is correct for our physical space?" The mathematical work revealed that this question must be answered by physical experimentation, not mathematical reasoning, and uncovered the reason why the experimentation must involve immense (interstellar, not earth-bound) distances. With the development of relativity theory in physics, this question became vastly more complicated.

Introduction of mathematical rigor

All the work related to the Parallel Postulate revealed that it was quite difficult for a geometer to separate his logical reasoning from his intuitive understanding of physical space, and, moreover, revealed the critical importance of doing so. Careful examination had uncovered some logical inadequacies in Euclid's reasoning, and some unstated geometric principles to which Euclid sometimes appealed. This critique paralleled the crisis occurring in calculus and analysis regarding the meaning of infinite processes such as convergence and continuity. In geometry, there was a clear need for a new set of axioms, which would be complete, and which in no way relied on pictures we draw or on our intuition of space. Such axioms, now known as Hilbert's axioms, were given by David Hilbert in 1894 in his dissertation Grundlagen der Geometrie (Foundations of Geometry).

Analysis situs, or topology

In the mid-18th century, it became apparent that certain progressions of mathematical reasoning recurred when similar ideas were studied on the number line, in two dimensions, and in three dimensions. Thus the general concept of a metric space was created so that the reasoning could be done in more generality, and then applied to special cases. This method of studying calculus- and analysis-related concepts came to be known as analysis situs, and later as topology. The important topics in this field were properties of more general figures, such as connectedness and boundaries, rather than properties like straightness, and precise equality of length and angle measurements, which had been the focus of Euclidean and non-Euclidean geometry. Topology soon became a separate field of major importance, rather than a sub-field of geometry or analysis.

Geometry of more than 3 dimensions

The 19th century saw the development of the general concept of Euclidean space by

regular convex polytopes in dimension four
, and three in all higher dimensions.

In 1878

quaternions with Hermann Grassmann
's algebra and revealing the geometric nature of these systems, especially in four dimensions. The operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modeled to new positions.

The 20th century

Developments in algebraic geometry included the study of curves and surfaces over finite fields as demonstrated by the works of among others André Weil, Alexander Grothendieck, and Jean-Pierre Serre as well as over the real or complex numbers. Finite geometry itself, the study of spaces with only finitely many points, found applications in coding theory and cryptography. With the advent of the computer, new disciplines such as computational geometry or digital geometry deal with geometric algorithms, discrete representations of geometric data, and so forth.

Timeline

See also

Notes

  1. The Bible
    , has been more widely used...."
  2. .
  3. ^ Eves, Chapter 2.
  4. ^ "Clay tablets reveal Babylonians discovered astronomical geometry 1,400 years before Europeans - The Washington Post". The Washington Post.
  5. ^ A. Seidenberg, 1978. The origin of mathematics. Archive for the history of Exact Sciences, vol 18.
  6. ^ (Staal 1999)
  7. ^ Most mathematical problems considered in the Śulba Sūtras spring from "a single theological requirement," that of constructing fire altars which have different shapes but occupy the same area. The altars were required to be constructed of five layers of burnt brick, with the further condition that each layer consist of 200 bricks and that no two adjacent layers have congruent arrangements of bricks. (Hayashi 2003, p. 118)
  8. ^ (Hayashi 2005, p. 363)
  9. ^ Pythagorean triples are triples of integers with the property: . Thus, , , etc.
  10. ^ (Cooke 2005, p. 198): "The arithmetic content of the Śulva Sūtras consists of rules for finding Pythagorean triples such as (3, 4, 5), (5, 12, 13), (8, 15, 17), and (12, 35, 37). It is not certain what practical use these arithmetic rules had. The best conjecture is that they were part of religious ritual. A Hindu home was required to have three fires burning at three different altars. The three altars were to be of different shapes, but all three were to have the same area. These conditions led to certain "Diophantine" problems, a particular case of which is the generation of Pythagorean triples, so as to make one square integer equal to the sum of two others."
  11. ^ (Cooke 2005, pp. 199–200): "The requirement of three altars of equal areas but different shapes would explain the interest in transformation of areas. Among other transformation of area problems the Hindus considered in particular the problem of squaring the circle. The Bodhayana Sutra states the converse problem of constructing a circle equal to a given square. The following approximate construction is given as the solution.... this result is only approximate. The authors, however, made no distinction between the two results. In terms that we can appreciate, this construction gives a value for π of 18 (3 − 22), which is about 3.088."
  12. ^ a b c (Joseph 2000, p. 229)
  13. ^ Mathematics Department, University of British Columbia, The Babylonian tabled Plimpton 322.
  14. ^ Three positive integers form a primitive Pythagorean triple if and if the highest common factor of is 1. In the particular Plimpton322 example, this means that and that the three numbers do not have any common factors. However some scholars have disputed the Pythagorean interpretation of this tablet; see Plimpton 322 for details.
  15. ^ a b (Dani 2003)
  16. ^ Cherowitzo, Bill. "What precisely was written over the door of Plato's Academy?" (PDF). www.math.ucdenver.edu/. Archived (PDF) from the original on 2013-06-25. Retrieved 8 April 2015.
  17. ^ "Archimedes". Encyclopedia Britannica.
  18. ^ Luciano Canfora; The Vanished Library; University of California Press, 1990. - google books
  19. ^ (Hayashi 2005, p. 371)
  20. ^ a b (Hayashi 2003, pp. 121–122)
  21. ^ (Stillwell 2004, p. 77)
  22. ^ Radha Charan Gupta [1977] "Parameshvara's rule for the circumradius of a cyclic quadrilateral", Historia Mathematica 4: 67–74
  23. ^ a b Needham, Volume 3, 91.
  24. ^ a b c Needham, Volume 3, 92.
  25. ^ Needham, Volume 3, 92-93.
  26. ^ Needham, Volume 3, 93.
  27. ^ Needham, Volume 3, 93-94.
  28. ^ Needham, Volume 3, 94.
  29. ^ Needham, Volume 3, 99.
  30. ^ Needham, Volume 3, 101.
  31. ^ Needham, Volume 3, 22.
  32. ^ Needham, Volume 3, 21.
  33. ^ Needham, Volume 3, 100.
  34. ^ a b c Needham, Volume 3, 98–99.
  35. ^ Needham, Volume 3, 98.
  36. S2CID 119868978
    .
  37. S2CID 10374218, archived from the original
    (PDF) on 2009-10-07.
  38. ^ Supplemental figures Archived 2009-03-26 at the Wayback Machine
  39. ^ d'Alverny, Marie-Thérèse. "Translations and Translators", in Robert L. Benson and Giles Constable, eds., Renaissance and Renewal in the Twelfth Century, 421–462. Cambridge: Harvard Univ. Pr., 1982, pp. 433–4.
  40. ^ M.-T. d'Alverny, "Translations and Translators," p. 435
  41. ^ Howard Saalman. Filippo Brunelleschi: The Buildings. (London: Zwemmer, 1993).
  42. ^ "...and these works (of perspective by Brunelleschi) were the means of arousing the minds of the other craftsmen, who afterwards devoted themselves to this with great zeal."
    Vasari's Lives of the Artists Chapter on Brunelleschi
  43. ^ "Messer Paolo dal Pozzo Toscanelli, having returned from his studies, invited Filippo with other friends to supper in a garden, and the discourse falling on mathematical subjects, Filippo formed a friendship with him and learned geometry from him."
    Vasarai's Lives of the Artists, Chapter on Brunelleschi
  44. ^ The Secret Language of the Renaissance - Richard Stemp

References

External links