Medieval vaulting geometry research, links, notes and keywords

Marcus Vitruvius Pollio(c. 80–70BCE, c. 15 BCE)

Mathes Roriczeralso Matthäus Roritzer, Röriczer (1435-1495)

Piero dellaFrancesa(*c.*1415–92)

Fra Luca Bartolomeo de Pacioli(1446 - 1517)

Leonardo di ser Piero daVinci(1452 – 1519)

Albrecht Durer(1471–1528)

Diego de Siloé(*c.*1490–1563)

Philibert de L’Orme (1514–70)

François Derand(1588or1591–1644)

Mathurin Jousse(1607–before1692)master locksmith

Gérard Desargues(1591–1661) my favorite

Amédée-François Frézier(1682–1773)

Gaspard Monge(1746–1818)

Charles-François-Antoine (1780-1854)

Mahan, D. H. (Dennis Hart), 1802-1871

Robert Willis (1800-1875)1843 gothic vaulting

*De Divina Proportione
*(1509)

Hans Lencker Perspectiva

Villard de Honnecourt and Euclidean Geometry -- Marie-Thérèse Zenner

Chancery building, Blois (Loir-et-Cher). 16th century. Philibert de l’Orme framework. Detail of the roofing: curved rafters and crown plates.

Philibert de l’Orme framework.

Vitruvius Scamilli -- Levelling Blocks

Cathedrals around: the Word geometric, Thierry Champris, Editions Guy Trédaniel

The object of dispute was whether the elevation of the cathedral of Milan was, from the plan already in place, be drawn by regulators'ad quadratum"(as the square and octagon) or"ad triangulum"(as the triangle and the hexagon star / seal of Solomon).And was not so much about technique as a token, by the importance given to numbers by the Neoplatonic doctrines.

http://www.google.com/url?sa=t&source=web&cd=28&ved=0CDkQFjAHOBQ&url=http%3A%2F%2Fwww.paperblog.fr%2F2172245%2Fautour-de-cathedrales-le-verbe-geometrique-par-thierry-de-champris-editions-guy-tredaniel%2F&ei=HwbMTPbyKY24sQOyopn1Dg&usg=AFQjCNEEx9q9xSeZyTseLPPZl5yvhJdg8Q&sig2=GOEWlpIuShJzby3r7xuU7g Cathedrals: The verb geometric[Paperback] AmazonFrance

Regiomontanus, Apian and Capuchin Sundials

Regiomontanus, Apian and Capuchin Sundials

Capuchin Friar

Analemmatic sundial by plotting abscissas and ordinates. Analemmatic Dial. Motto: The Light and Shadow of God

Analemmatic sundials: How to build one and why they work

The analemmatic sundial is based on an ellipse which is marked out in hour points

An ellipse is an example of a curve called a conic section. Other conic sections are the circle, the parabola and the hyperbola. The Earth travels around the sun on an elliptical path.

Analemmatic sundials

umbra recta: Latin for "upright shadow", it is the label often found on the cotangent scale of altitudes < 45º on a shadow square.

umbra versa: Latin for "reverse shadow", it is the label often found on the tangent scale of altitudes > 45º on a shadow square.

Basic sundial nomenclature

umbra recta -- the shadow cast by a horizontal stick on a vertical surface

umbra versa -- the shadow cast by a vertical stick on a horizontal surface

Quadrato geometrico per misurare ogni lunghezza (geometrical square to measure every length) 60

Scala altimetra di gradi 100 per misurare ogni altezza da lontano (Altitude scale of 100 degrees, to measure every far height) 12 .

"Ecliptic" simply means "(oblique) circle" (in Greek ekklitiké, from klinein = to bend)

Claudius Ptolemaeus (Ptolemy) epicycles

Claudius Ptolemaeus (Ptolemy) Apotelesmatika -Quadripartitum (four books) of astrological treatise

CHRONOLOGY – some selected dates in the development of sundials and solar astronomy

development of sundials and solar astronomy

Analemmatic Dial. Motto: The Light and Shadow of God

Analemmatic sundial by plotting abscissas and ordinates. The point (3,2) has 3 as its abscissa (x) and 2 as its ordinate (y).

Drive the chariot of the sun’s arc south of the equator to see the chariot of light at winter solstice.

Hexagram

mandala , [Skt.,=circular, round] a concentric diagram , Sacred Islamic Geometry grows from a circle

star of David or Solomon's seal

Treatise On Those Parts of Geometry Needed by Craftsmen PDF files

A lesson in Applied Geometry and Euclidean Geometry

String Theory on Medieval Masons and Carpenters drawing an Ellipse

π

Nexus Network Journal 9,2: Architecture and Mathematics

Oval Domes: History, Geometry and Mechanics Santiago Huerta

Oval Domes: History, Geometry and Mechanics Santiago Huerta

Sloan's constructive architecture : a guide to the practical builder and
mechanic (1859)

4 rectangles of crown moulding design-layout page 232

"In which is contained a series of designs for domes, roofs and spires, with
a number of plates showing the interior construction and finish of bays, window
shutters, sliding doors, etc., designed expressly for the joiner's use; choice
examples of the five orders of architecture, selected from the most celebrated
specimens of antiquity, with the figured dimensions of their height, projection
and profile, and their division into parts; to which is added a number of
useful geometrical problems, examples of groins, centering for arches, diagrams
of stair lines, with architraves, door moldings, etc.; the whole being illustrated
by sixty-six carefully prepared plates, accompanied by explanatory text and
general essays, to which is appended a copious glossary"--T.p

Sloan's
constructive architecture : a guide to the practical builder and mechanic

Encyclopedia of architecture : a dictionary of the science and practice of architecture, building, Carpentry, etc., from the earliest ages to the present time, forming a comprehensive work of reference for the use of architects, builders, carpenters, masons, engineers, students, professional men, and amateurs ([185-?])

Volume 2

Peter Nicholson ( 1765-1844) talking about Vitruvius

Peter Nicholson talking about Vitruvius

Peter Nicholson talking about Brunelleschi

Peter Nicholson talking about sima recta crown moulding design

Peter Nicholson talking about panormara projection

Peter Nicholson talking about panormara projection

Peter Nicholson talking about stereotomy spherical Triangles spherical Trigonometry

V-1 Peter Nicholson talking about Bracketing

V-1 Peter Nicholson talking about Double Curveture Arches

V-1 Peter Nicholson talking about Conic Sections

V-1 Peter Nicholson talking about Cylinders

V-1 Peter Nicholson talking about Domes

V-1 Peter Nicholson talking about Ovals and Ellipses

V-1 Peter Nicholson talking about Hip Roof Geometry

Solid Geometry for Builders .. Hoppers Practical geometry for builders and architects 1921

Cyclopedia Of Architecture, Carpentry, And Building Vol7-10

Cyclopedia Of Architecture, Carpentry, And Building Vol7-10

The encyclopædia britannica Vaulting 1911

Mouldings of the Tudor period : a portfolio of full size sections-- Small, Tunstall

The Standard moulding book 1920

Guide for drawing the acanthus, and every description of ornamental foliage

Sloan's constructive architecture : a guide to the practical builder and mechanic. In which is contained a series of designs for domes, roofs and spires, with a number of plates showing the interior construction and finish of bays, window shutters, sliding doors, etc., designed expressly for the joiners use; choice examples of the five orders of architecture, selected fromthe most celebrated specimens of antiquity, with the figured dimensions of their height, projection and profile, and their division into parts,; to which is added a number of useful geometrical problems, examples of groins, centering for arches, diagrams of stair lines, with architraves, door mouldings, etc., the whole being accompanied by explanatory text and general essays, to which is appended a copious glossay

Sloan's constructive architecture

Roof framing made easy: a practical and easily comprehended system, adapted ... By Owen Bernard Maginnis

Roof framing made easy 1896 Maginnis, Owen B. (Owen Bernard), b.

Egypt, Greece, Roman, France, Islamic

http://www.en.charpentiers.culture.fr/treesintohouses/thecarpentersart/specialistliterature

Technical knowledge about wood and framing appear in specialist literature that was particularly abundant between the 16th and 19th centuries.

https://www-docs.tu-cottbus.de/bautechnikgeschichte/public/openaccess/negre_oa.pdf

Some Considerations on Traité de L’Art de Bâtir by Rondelet
and the

Technical Literature of his Time

The contemporary world of building is dominated by standardization and mechanization that are transforming builders into mere labourers devoid of any creative thinking.

http://www.google.com/url?sa=t&source=web&cd=1&ved=0CBYQFjAA&url=http%3A%2F%2Fwww.unesco.org%2Fculture%2Fich%2Fdoc%2Fsrc%2F00251-Nomination_form.doc&rct=j&q=Nicolas%20Fourneau%20french%20carpenter&ei=LZjmTND8No7GsAO-2biwCw&usg=AFQjCNHKvkcTVLoEnnlV_27t_esxyU9JKg&sig2=S2N7zqGj2vFUpfbCp6slew&cad=rjt

The scribe tradition in French timber framing, or the “trait de charpente” as the carpenters call it, makes it possible to design complex wooden buildings in three dimensions.

The symbolic aspects of instruction in the French scribe method are revealed in the context of the Compagnonnage only in a confidential manner during joining and induction ceremonies. These very particular circumstances are not open to outsiders, nor are they communicated to the outside world in any way. However, the general principles of this instruction, the universal aspects of which are affirmed by the Companions themselves, can be disseminated without restraint. The technical and historic aspects of scribing, the production of masterworks and masterpieces at the end of apprenticeships may be referred to normally, taking into account, of course, the great complexity of some productions.

LaHire, Philippe de, La gnomonique ou methodes universelles, pour tracer des horloges solaires ou cadrans sur toutes sortes de surfaces , 1698

Gabriel-Philippe de LaHire (1677-1719), in mathematics

Mathurin Jousse's work, published in 1627, reprinted and supplemented by
Gabriel Philippe de La Hire in 1702, illustrating the implementation of various
framing attic, timber framed or stairs, is the first expression.

http://architectura.cesr.univ-tours.fr/Traite/Notice/CESR_IHA615.asp

http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?pn=336&url=%2Fmpiwg%2Fonline%2Fpermanent%2Flibrary%2FUSVW0U1V%2Fpageimg&start=331&viewMode=images&mode=imagepath

http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?pn=340&url=%2Fmpiwg%2Fonline%2Fpermanent%2Flibrary%2FUSVW0U1V%2Fpageimg&start=331&viewMode=images&mode=imagepath

Stereotomy a multifaceted technique

Descriptive geometry, belongs to the history of techniques through its origins,
While stereotomy, together with carpentry, provides

one of the richest examp]es of the uses of applied geometry, it is also at
the root of a branch of erudite geometry, namely descriptive geometry.

Before the 18th century, builders only had extremely simple, purely geometric
and (at best) empirical «rules» at their disposal to size the
building s under construction. One of the most famous rules is the «Leonardo
rule», which says the arch will not break if the chord of the

outer arc does not touch the inner arc.

Descriptive Geometry == «theoretical stereotomy», detached from its original function as a technique of stone cutting

Sand Geometry

Light Geometry

Egyptian Geometry

Greek Geometry

Roman Geometry

Arabic/Islamic Geometry

Persian Geometry

Pythagoras Geometry

Euclidean Geometry

Vitruvian Geometry

Archimedes Geometry

Apollonios Geometry

Sacred Geometry

Vesica Piscis Geometry

Ad Triangulum Geometry

√3 Geometry

Ad Quadratum Geometry

√2 Geometry

Euclid Elements

Euclidean postulates

Vitrúvio

Vitruvius

Vitruvian principles

Symmetry Arises from Proportion

A cubit is a forearm, from the elbow to the tip of the fingers.

analemmas

dos princípios vitruvianos

Ad Triangulum

Ad Quadratum

Ad Quadratum, Ad Triangulum and The Sacred Cut

The Vesica Piscis and Squaring of the Circle

Ad Triangulum ( triangle within the circle)

Ad Triangulum (hexagonal base)

Ad Triangulum (√3 base)

Ad Triangulum (three point geometry)

Ad Triangulum
(six point geometry)

Ad Triangulum
(human consciousness)

Ad Triangulum (Heaven)

Ad Quadratum (square within the circle)

Ad Quadratum (octagonal based)

Ad Quadratum (√ 2 based)

Ad Quadratum (four point geometry)

Ad Quadratum
(eight point geometry)

Ad Quadratum
(earth geometry)

Ad Quadratum (physical world)

Ad Quadratum -- Height =( width * √ 2)
/ 2

Ad Triangulum -- Height = width * (√3 /
2)

Vitruvius wrote that if a figure of a correctly proportioned human body were
placed within a square (homo ad quadratum), which in turn was placed in a
circle (homo ad circulum) in such a way that the corners of the square were
just touching the arc of the circle, then the precise centre of both the circle
and the square would be the human figure's navel (umbilicus ad circulum et
ad quadratum). <\p>

2* π * height = perimeter

√2 = 1.414

Golden Ratio phi = (√5 + 1) ÷ 2 ) = 1.6180339887

Euclid -- extreme and mean ratio

Luca Pacioli --"divine proportion" in Divina Proportione

Egyptian pyramids base (b)= 1, √phi = height(h), phi = apothem(a), slope angle = arctan(√phi ) = 51.83°

1:1.414

12 times 1.414 is 16.97”

"Without symmetry and proportion there can be no principles in the design
of any temple; that is, if there is no precise relation between its members,
as in the case of those of a well shaped man."

Vitruvius, Book III, Chap. 1

*Six petalled rose*

Sun-rosette

Daisy wheel

Flower of Life

Seed of Life

Morning Star

Dutch Hexagram

Vesica Piscis

Thunder Bolt Marks

Tripod of Life

Ancient Egyptian Seal of Solomon

6-pointed star

SIX TRIANGLES...is the Egyptian hieroglyphic for the ...Land of the Spirits.

6-pointed star was the first sign or hieroglyphic of Amsu

STAR OF YOUR GOD

Light Geometry

Romania geometry thunder rosette

carpenter geometry thunder rosette

"Halloween...emanates from the 14th century... when the Druids would knock
on the doors of the castles demanding the young maiden or princess for their
sacrifices. If they were not given the maiden, they would paint a HEXAGRAM
on the door to tell...that all should die in that household."

8 pointed star , star of the dawn or MORNING STAR'

The Menorah, not the Hexagram is the true symbol of God's covenent with the Jewish people.

Many Jews and Christians have been deceived by Jewish Kabbalists who would have them believe that the six-pointed star is a Jewish symbol. Nothing could be further from the truth. It is not a Jewish symbol, but an occult symbol. The six-pointed star is a hexagram – a curse mark – no matter what name it may have: the Star of David, Solomon’s Seal, Double Triangle, Shield of David, etc. When the occult practitioner puts a curse on someone, he uses the hexagram (a “Hex!”)

“The Universal Jewish Encyclopedia declares that the SIX-POINTED STAR…according to the Rosicrucians…was known to the ancient Egyptians.” “SIX TRIANGLES…is the Egyptian hieroglyphic for the …Land of the Spirits.” “Ancient Egyptian Seal of Solomon” “In the Astro-Mythology of the Egyptians, we find belief in the first man-god (Horus I) …and his death and resurrection as Amsu” “This (6-pointed star) was the first sign or hieroglyphic of Amsu” “Amsu – the risen Horus – was the first man-god risen in spiritual form.”

Geometry is the greek word for earth measure, using angles with predictably moving stars to measure the earth in ancient times, as the Greek unit of length, the stade, is 600 greek feet (of 12.16 modern inches).

Medieval baptismal font

Gothic architecture

Gothic history

Gothic Mason’s Marks

Voutes

Vault

Ribbed Cross vault diagonals

Ellipse

Oval

intersection horizontal vertical cylinder

Cone

Cylinder

Sphere

Platonic Solids

Intrados

Cylindroid

Spheroid

Spherical elliptic geometry

De Charpentier

Carpenter

Helicoidallywarped = skewed arch

conoids, hyperboloid arches

Trumpet arch

Warped surfaces arches of double curvature.

A obra, intitulada On the
Ordination of Pinacles, Röriczer

Canônico

Canonical

Speculative Masons

maçons especulativos

chronological list of some of the most important mathematicians in history

The Story of Mathematics

Pythagoras
of Samos (c. 570-c. 495 BC)

Plato(c. 428/427 BC– 348/347 BC)

Archytas of Tarentumwas(428–347 BC) Archytas Curve--determining a certain
point as the intersection of three surfaces of revolution, (1) a right cone,
(2) a cylinder, (3) a tore

Euclid(300
BC)

Archimedes
of Syracuse(c*.*287BC –*c.*212BC)

Apollonius
of Perga[Pergaeus]
(ca. 262 BC–ca. 190 BC)

Claudius Ptolemy (2nd C. C.E.)

The oldest known work on trigonometric tables is the Syntaxis Mathematica written by Ptolemy of Alexandria about 140 AD.

The Theorem of Ptolemy

According to the theorem of Ptolemy, if a quadrilateral is inscribed in a circle, the sum of the products of the two opposed sides is equal to the product of the multiplication of the diagonals.

According to the quadrilateral PQRS, we have the following formula.

PR x QS = PQ x RS + PS x QR.

The theorem of Ptolemy is as follows:

c2 (a) = 60 c2 (2 a) / 120 + c (180 – 2 a)

Marcus Vitruvius Pollio(c. 80–70BCE, c. 15 BCE)

Mathes Roriczeralso Matthäus Roritzer, Röriczer (1435-1495)

Piero dellaFrancesa(

Fra Luca Bartolomeo de Pacioli(1446 - 1517)

Leonardo di ser Piero daVinci(1452 – 1519)

Albrecht Durer(1471–1528)

Diego de Siloé(

Philibert de L’Orme (1514–70)

François Derand(1588or1591–1644)

Mathurin Jousse(1607–before1692)master locksmith

Gérard Desargues(1591–1661) my favorite

Amédée-François Frézier(1682–1773)

Gaspard Monge(1746–1818)

Charles-François-Antoine (1780-1854)

Mahan, D. H. (Dennis Hart), 1802-1871

Robert Willis (1800-1875)1843 gothic vaulting

Fra Luca Pacioli.

Leonardo da Vinci

Stereotomy

Stereography

Stereotomic

Stereotomique

Stereotomic treatises

Skewed arches,trompes

Stone cutting

Complex stereotomic cuts

Stereotomically complex cuts

Traite de stereotomie

Traite Descriptive Geometry

Stereotomic descriptive geometry

Traite de stereotomic descriptive geometry

Traite de stereotomy descriptive geometry

Companions of the Tour de France

Vault crossed with ribs

Pendentive

Cistercian monasteries

Euclid Conic sections

Canted elliptical arcs

elliptical half-cylinder

golden number is (1 + √5)/2
= 1.618033989

Great Pyramid rise at an angle of 51° 52'.

Vault descriptive geometry INTERSECTIONS OF CURVED SURFACES

Medieval building techniques

Valknut symbol

gotland churchs

Anahata chakra

Symbolizes the consciousness of love, empathy, selflessness and devotion.
On the psychic level, this center of force inspires the human being to love,
be compassionate, altruistic, devoted and to accept the things that happen
in a divine way.

Square oven *four*, spandrel

Church

Cathedral

Spherical vault

Voussoirs

Freemasonry

Tectonic

Tekton

Orthographic projections

Mediaeval practical geometry

Plane Geometry

Centring

Centering

Trammel of Archimedes

An ellipsograph is a trammel of Archimedes intended to draw, cut, or machine
ellipses, e.g. in wood or other sheet materials.

J. W. Downs: Practical Conic Sections: The Geometric Properties of Ellipses,
Parabolas and Hyperbolas.

Frustum of a Cone

Walk as children of light

Oculus(pluraloculi) is the Latin word foreye

Author (s) Jousse, Mathurin

Title The scene of the carpenter's art ...Title The secret architecture .

Address The Arrow, George Griveau, 1627

Location Paris, BENSBA Reserve SLE. 1250

Words matter Carpentry, Orders

Mathurin Jousse is best known for having written between 1627 and 1642, three treaties building devoted to the locksmith, carpentry and stereotomy, The faithful initiation of the locksmith art, theater art of carpentry and Secrecy architecture.

http://architectura.cesr.univ-tours.fr/traite/Notice/ENSBA_LES1250.asp

http://www.insitu.culture.fr/article.xsp?numero=1&id_article=flb001-857

Author (s) Derand, François

Title The architecture of arches ...

Address Paris, Sebastien Cramoisy, 1643

Location Besancon, City Library, 11598

Words matter Stereotomy, vaults

Derand Francis was born in Vic-sur-Seille in the diocese of Metz between 1588 and 1591.

http://architectura.cesr.univ-tours.fr/traite/Notice/B250566101_11598.asp

Author (s) Androuet Hoop, Jacques

Title Quinque and viginti arcuum exempla ...

Address Orleans, sn, 1549

Location Paris, Binh, 4 RES 1475

Words matter Arcs Entries

http://architectura.cesr.univ-tours.fr/traite/Notice/B250566101_11598.asp

Author (s) Dumb, Pierre

Title Traict five orders of architecture, which have seruy elders. Translated
from Palladio Increased tidings of such device for the art of well built ...

Address Paris, Francois Langlois, 1645

Location Paris, BENSBA THE 646

Words matter Carpentry, Orders, Door Gambrel Roof

http://architectura.cesr.univ-tours.fr/traite/Notice/B250566101_11598.asp

Roof Truss within Arches

http://architectura.cesr.univ-tours.fr/traite/Images/LES646/ENSBA_LES646_247.jpg

The Dutch Gambrel is taken from this reprint by Palladio that the Dutch Stole

http://architectura.cesr.univ-tours.fr/traite/Images/LES646/ENSBA_LES646_258.jpg

François Mansart (13 January 1598 in Paris - 23 September 1666 in Paris) was born to a master carpenter

counterfeit French Mansart Roof --> Dutch Gambrel Roof

published in France 1645, counterfeit Palladio published in Amsterdam in 1646

http://architectura.cesr.univ-tours.fr/traite/Notice/ENSBA_LES646.asp?param=

This partial translation of Palladio, Book I, enjoyed a second printing two years later, in 1647, at Pierre Mariette. Three counterfeit Dutch successively published in Amsterdam in 1646, 1679 and 1682 attest to its success. In 1646 the book was translated into Dutch, and was also published in London in 1663 and continually reprinted. The fortunes of Palladio The Mute in northern Europe is likely due to its particularity: revised and corrected in the French style, this version was better suited to the Italian edition of practitioners whose ideas and techniques, climate forcing, were close.

Carpentry

L'Orme, Philibert (1514-1570)

New inventions for ... well built, Paris, Frédéric Morel, 1561.

New inventions for ... well built , Paris, and William Jerome Marnef Cavellat,
1576

Jousse, Mathurin

The art of theater carpenter enriched various figures ... , La Fleche, George
Griveau, 1627.

The art of theater carpenter enriched various figures ... , La Fleche, George
Griveau, 1650.

The scene of the carpenter's art, various figures ... enriched in La Fleche,
Widow of George Griveau, 1659.

The art of theater carpenter enriched various figures , La Fleche, Widow of
George Griveau, 1664.

The art of carpentry ... Fixed & increased ... By Mr. DLH .. , Paris,
Moet Thomas, 1702.

Dumb, Pierre (1591-1669)

Basti way for all kinds of people , Paris, Melchior Tavernier, 1623

Manner of well built for all sorts of people , Paris, Francois Langlois, 1647

Manner of well built for all sorts of people ... ; Increase new ... bastimens,
Paris, Jean du Can, 1663.

Treaty of Palladio

Traict five orders of architecture ... Translated from Palladio increased
tidings inventions for the art of bastir by Sr. The Mute , Paris, Francois
Langlois, 1645.

Stereotomy

Blondel, Francois (1618-1686)

Four problems

Resolution of the four main problems of architecture , Paris, Imprimerie Royale,
1673.

Derand, Francis (1588? -1644)

The architecture of arches, or the art features and cutting vaults ... , Paris,
Sebastien Cramoisy, 1643.

Desargues, Girard

Practice relating to evidence, Mr Desargues Lyonnois, to cut stones in architecture
... , Paris, Pierre Deshayes, 1643.

L'Orme, Philibert (1514-1570)

The first volume of the architecture ... , Paris, Frédéric Morel,
1567.

The architecture ... , Paris, and William Jerome Marnef Cavellat, 1576.

The architecture ... , Paris, Guillaume Auvray, 1603.

... Full implementation architecture containing Unze Books, plus two ... ,
Paris, Regnault Chaudière II, 1626.

... Full implementation architecture containing Unze Books, plus two ... ,
Rouen, David Ferrand, 1648.

Jousse, Mathurin

The Secret ... architecture , La Fleche, George Griveau, 1642.

Perrault, Charles (1628-1703)

Parallel

Parallel of the Ancients and the Moderns, which looks into the arts and sciences.
... Dialogues , Paris, Jean-Baptiste Coignard, 1688 (Volume 1).

Antonio da Sangallo the Younger

http://www.all-art.org/history230-7-1.html

The Architectural Drawings of Antonio da Sangallo the Younger and His Circle

Volume 1 - Fortifications, Machines, and Festive Architecture

Hans Lencker Perspectiva

Squaring the Circle: Geometry in Art and Architecture

Calter, Paul A., Vermont Technical College

Villard de Honnecourt and Euclidean Geometry -- Marie-Thérèse Zenner

Master carpenters who coped with Greek revival styles during the 18th and early 19th centuries employed a variety of constructive methods to create the ornate curves these styles demanded. Handbooks like Asher Benjamin’s The American Builder’s Companion compiled these techniques for carpenters to reference.

Drawing Hour Lines with Elliptical Coordinates

The American builder's companion : or, A system of architecture, particularly adapted to the present style of building ; illustrated with seventy copperplate engravings (1827)

Sacred Stones project of the Abbey of New Clairvaux

http://www.sacredstones.org/

Sacred Stones Chapter House Construction, 2010

http://picasaweb.google.com/106628474194260084428/ChapterHouseConstruction2010#

Ad Quadratum, the Sacred Cut, & Roman Architecture

http://www.dartmouth.edu/~matc/math5.geometry/unit5/unit5.html

LES LIVRES D’ARCHITECTURE, Derand Francis, Stereotomy, vaults, Trammel for ellipse

Philibert De l’Orme
as early as 1561, in the Nouvelles inventions

Philibert De l’Orme describes how to use ordinates to draw the desired curve, but calls it **curves lengthened** and referrs to it as softening the curve.

All of Philibert De l’Orme ,1515, books at Open Archive

All of Mathurin b. Jousse, 1607, books at Open Archive

http://architectura.cesr.univ-tours.fr/Traite/Notice/ENSBA_Les1653.asp?param=en

Philibert De l'Orme septentrion boreas

As we can trace the foundations of a building by means of a perpendicular at the end of a straight line.
CHAPTER I.

The invention to the bracket by means of a triangle came from Pythagoras, and he sees himself in the ninth book of Vitruvius, Chapter II. And serves as the triangle of lines and proportions only to that square, but also for many other things and other figures and geometrical instruments necessary and required to help construct buildings, and measure, I will not say the area, but all heights and widths, as I will show when he comes about.

Take the case that you have drawn the line QR, and hereunto an equilateral triangle, that is to say as great on one side than the other, as you see RST, T is the point where you pull another curved line marked Z, is tightened without moving the compass, and requires that the distance ST is similar to that of TZ. This makes you draw a straight line from point S to T, till it intersects the line Z, and this place, as you see the point marked X, you draw another line even to the point of R, which will be precisely perpendicular to the line QR, as you can judge by the figure followed.

In the first picture above, of the first speed square, Philibert De l'Orme
developed it like a sundail.

French to English translation

LE PREMIER TOME DE L'ARCHITECTURE DE PHILIBERT DE L'ORME, 1567

Be given an **equilateral triangle **such width as you like,
like ABC, the more it grows, so will the insurance and kindness. Where do
I did not wish to help more than that which you see below figuratively, by
as much as I used to be easier in my coffers, and do not usually point thereof,
a **Astrolable**

be, and ephemeris, with a few other books, and cases filled compass, and what
it takes to portraire. Within this triangle imagine a circle, as you can see
marked EFGH (**almost as if it were a dial showing the hours**)
and divide into so many parts that will, like **twenty-four**,
thirty-two, forty eight, the most that there is the best.

I divided the latter into thirty-two, and is set amidst a magnetic needle,
as well as marine dials and compasses, or small whom we help to find the**
hours to the Sun**, but notice that said needle must be very good and
very moving. When you want to help the triangle, you look through one side
as you please, for the one marked in Figure D. This makes you discard your
city view, castle or place from which you want to take the form and figure,
and make a sketch first on paper scored coarsely and you can understand the
decision. Can you make the trip at all. If you want that it should fit in
memory or writing a bend and each face of the walls to measure the length
as you will see below. Having done this, you can start at one end of the castle,
town or place, keeping your triangle against the first section of wall with
a ruler to have greater decision, against which must be your triangle as you
see marked K. This makes you look at where does the needle, and how many,
if ten or some other number, whatever it is, you mark the outline of your
paper, right place cons which you presented your triangle. And after you go
to another detour section of wall, and do as you did, with the rule and said
your triangle against that section of wall, and looking at the number that
stops the tip of the needle, which you also put on a sketch that was made
to place all as before, and continue in this way all enceinture and around
the city, or other place, but still marking to each piece of wall and bend
the numbers on which the needle stop of your triangle, as I said, in a similar
length that each contain a section of wall. All this well-ordered, when you
want to make a fair level of your town or castle, you will extend over a table
on paper or parchment on which you want portraire, being well bonded and attached
around the edges. But you'll have the table farm, and can not turn here then
at least just that this, that all lines of turn are drawn. Then you look at
how many yards to your city in length and width and if there are a hundred
or two hundred, we will divide into many parts numbers and the length of your
paper, reducing them to small yards, for which you any measures to give your
design that starts at one end, where to put the triangle where you are helped,
and turn until the needle is exactly the number it was when you presented
against the wall of the city. But do remember to put the lengths that you
found each piece of a wall on their own place. This makes you pull the line
throughout the side of your triangle D, which said triangle has been presented.
For the better such a triangle must be made of any material that is not thick,
like copper, brass, silver, or wood that hairline, so that it can easily take
the above line, and has been accustomed to do with a ruler. You will present
to the said triangle and to complete all other sections of wall, and turn
till the needle is located exactly on the number to which it was when it was
presented cons

Chancery building, Blois (Loir-et-Cher). 16th century. Philibert de l’Orme framework. Detail of the roofing: curved rafters and crown plates.

Philibert de l’Orme framework.

Drawings of Leonardo da Vinci

Geometrical construction of the Vitruvian Man by Leonardo da Vinci

Study of proportions, from Vitruvius's De Architectura , Vitruvian man

Studies of Concave Mirrors of Differing Curvatures, c. 1492

Rib vault From Wikipedia

http://en.wikipedia.org/wiki/Rib_vault

Groin vault From Wikipedia

http://en.wikipedia.org/wiki/Groin_vault

Centring From Wikipedia

http://en.wikipedia.org/wiki/Centring

German vaulting From German Wikipedia

http://de.wikipedia.org/wiki/Gew%C3%B6lbe

German rib vaulting From German Wikipedia

http://de.wikipedia.org/wiki/Gew%C3%B6lberippe

List of architectural vaults From Wikipedia

http://en.wikipedia.org/wiki/List_of_architectural_vaults

List of arch types From Wikipedia

http://en.wikipedia.org/wiki/Arch

http://www.artemisillustration.com/assets/text/Ornamental%20%20Terms_Glossary.htm

Hexagramm From German Wikipedia

http://de.wikipedia.org/wiki/Hexagramm

Peterborough Cathedral-The Complete Geometry 1100 – 1500

Canterbury Cathedral by COLIN JOSEPH DUDLEY

http://medievalarchitecturalgeometry.com/A3%20The%20Geometry.htm

Ad Triangulum (And maybe some Ad Quadratum too)

Timber Framers Guild

http://www.tfguild.org/forums/ubbthreads.php?ubb=showthreaded&Number=24132&PHPSESSID=9b674f8192c18c2d4f574afb5812f6d1

Designing With the Daisy Wheel

Timber Framers Guild

http://www.tfguild.org/forums/ubbthreads.php?ubb=showthreaded&Number=23838&PHPSESSID=855089735f286dc0b4aae183f8793dbc

Daisy wheel geometry

http://nicolawood.typepad.co.uk/kesurokai/daisy-wheel-geometry/

Designing with the daisy wheel

http://www.timberstructures.net/timber-framing.html

Doubling the Circle - Sacred Geometry

http://dcsymbols.com/double_circle/double.htm

**Nexus Network Journal**

Pointed Arches

This gave inspiration of the static qualities of the equilateral triangle,
well tested in Muslim architecture, and inspired the practical geometry (construction
geometry) of Gothic Europe. The best pointed arch (with an inscribed equilateral
triangle) has the geometry of the egg, which, being ruled by the Golden Number
(accompanied by the number 3), determines a form that meets such conditions
due to its being a system of great stability because of the harmony between
its parts. The use in architecture of the geometric regularity of the bird's
egg, in its paradigmatic form, besides satisfying aesthetical, constructive
and economic conditions, allows the thrusts to be transmitted to the ground
more directly and with minimal lateral efforts.

Arabic Euclidian architecture Pointed Arches

**Nexus Network Journal**

Didactics: Proportions in the Architecture Curriculum

Abstract.Roger Herz-Fischler presents a revised version
of a chapter entitled "Proportions" that appeared in the problems
part of his book,Space, Shape and Form /An Algorithmic Approach, developed for
a mathematics course he taught in the School of Architecture at Carleton University
from 1973-1984.

http://www.emis.ams.org/journals/NNJ/Didactics-RHF.html

**Nexus Network Journal **

Ad Triangulum

Transversal section of the Cathedral (Duomo) of Milan,
from Cesariano's 1521 edition of Vitruvius (Plate
LVI)

http://www.emis.ams.org/journals/NNJ/RHF-fig07.html

**Nexus Network Journal**

SPIRALS FROM THE GOLDEN RECTANGLE

Abstract.John Sharp examines spirals and the Golden Section for the Nexus Network
Journal, Winter 2002.

http://www.emis.ams.org/journals/NNJ/Sharp_v4n1-pt02.html

**Nexus Network Journal**

The Geometer's Angle No. 6: R-Tiles

AbstractThe square root of the golden section, root-phi,
is at times not given its due in discussions regarding the golden section.
This is an unfortunate situation because the geometric properties of provide
us with a unique and rarely applied design tool. In this column, geometer
Mark Reynolds focuses on the root-phi rectangle and its abilities to create
an exciting tiling, R-Tiles.

http://www.emis.ams.org/journals/NNJ/GA-v4n1-MAR.html

**Nexus Network Journal** 9,2: Architecture
and Mathematics

By Kim Williams

http://books.google.com/books?id=pMvG5_H5wDUC&lpg=PA226&ots=LXe8-bIXQb&dq=The%20Book%20of%20Architecture%20of%20Hern%C3%A1n%20Ruiz&pg=PA228#v=onepage&q=The%20Book%20of%20Architecture%20of%20Hern%C3%A1n%20Ruiz&f=false

**Nexus Network Journal** About the ellipse:

It is true that ancient Greeks called the conics stereoi topoi (solid loci). However, they knew well the plane properties of these curves ( boast of Apollonius of Perga, 3rd-2th century B.C., but already partially known to Menecmo) . They did not have the mathematical tools to calculating the perimeter of the ellipse exactly (i.e., the elliptic integrals introduced in the works of Euler and Legendre ) but they could calculate the area of the ellipse (Archimedes, On Conoids and Spheroids, prop. 6, its approximation depending only on p ) . Apollonius's work was known during the Roman imperial age. The discovery of the gardener's method, namely a method to trace an ellipse by means of a rope string and two pivots, is attributed to Anthemius of Tralles (mathematician and one of the architects of Sancta Sophia). One can always suppose that this method is far older: according to some researchers it dates back to Neolithic age.

Why are Roman amphitheatres elliptical?

Vitruvius Scamilli -- Levelling Blocks

Translated and read with passion by the Renaissance humanists, the teaching of De architectura will overshadow much of the expertise of the builders of the Gothic period.

While the Romanesque liked the symbolism of numbers that referred directly to the biblical message and its interpretation, Gothic builders revere above all the geometry.

Article published in theEncyclopaedia Universalis, Corpus, January 1993, vol 2, pp. 843-851)

Joel Herschman - The Wise Master Builder: Platonic Geometry in Plans of Medieval Abbeys and Cathedrals

In all probability the Lombards are the originators of this device so pregnant
of future possibilities. The new vault, groined, ribbed, and domed, was in
a class by itself, apart from anything that had gone before. Particularly
did it differ from the Roman vault in that, while the latter had a level crown,
obtained by using semicircular lateral and transverse arches and elliptical
groin arches (naturally formed by the intersection of two semicircular barrel
vaults of equal radius), the **"Lombard"** vault was constructed
with semicircular diagonals, the result being that domical form which was
always retained by the Gothic builders of France because of its intrinsic
beauty. Finally, the new diagonals suggested new vertical supports in the
angles of the pier, and so we obtain the fully developed compound pier, which
later, at the hands of the English, was to be carried to such extremes of
beauty, and to form a potent factor in the development of the Gothic structural
system.

Gothic Architecture

ANCIENT-WISDOM.CO.UK Sacred Geometry

http://www.ancient-wisdom.co.uk/sacredgeometry.htm

Cathedrals around: the Word geometric, Thierry Champris, Editions Guy Trédaniel

The object of dispute was whether the elevation of the cathedral of Milan was, from the plan already in place, be drawn by regulators'ad quadratum"(as the square and octagon) or"ad triangulum"(as the triangle and the hexagon star / seal of Solomon).And was not so much about technique as a token, by the importance given to numbers by the Neoplatonic doctrines.

http://www.google.com/url?sa=t&source=web&cd=28&ved=0CDkQFjAHOBQ&url=http%3A%2F%2Fwww.paperblog.fr%2F2172245%2Fautour-de-cathedrales-le-verbe-geometrique-par-thierry-de-champris-editions-guy-tredaniel%2F&ei=HwbMTPbyKY24sQOyopn1Dg&usg=AFQjCNEEx9q9xSeZyTseLPPZl5yvhJdg8Q&sig2=GOEWlpIuShJzby3r7xuU7g Cathedrals: The verb geometric[Paperback] Amazon

Price: £ 203.30

http://translate.googleusercontent.com/translate_c?hl=en&sl=fr&u=http://www.amazon.fr/gp/product/2844455565%3Fie%3DUTF8%26tag%3Dcompagnetcomp-21%26linkCode%3Das2%26camp%3D1642%26creative%3D6746%26creativeASIN%3D2844455565&prev=/search%3Fq%3DAd%2BTriangulum%26start%3D20%26hl%3Den%26newwindow%3D1%26sa%3DN%26rlz%3D1C1GGGE_enUS400US400%26prmd%3Divb&rurl=translate.google.com&twu=1&usg=ALkJrhgowz2F2eXiKUOSQtFE0VAt45rjNg

This looks like a good book, but 203 euros?

L e Compagnonnage

More about the book at compagnonnage.info Stereotomique

www.compagnonnage.info| THE SITE OF RESEARCH OF
JEAN-MICHEL MATHONIÈRE AND CENTRAL STUDY OF GUILDS

Cathedrals around: the Word geometric, Thierry Champris, Guy Trédaniel editions.

August 31, 2009

Cathedrals
around: the Word geometric, Thierry Champris, Guy
Trédaniel editions.

http://translate.google.com/translate?hl=en&sl=fr&u=http://compagnonnage.info/blog/blogs/blog1.php/2009/08/31/autour-de-cathedralesn-le-verbe-geometri&ei=HwbMTPbyKY24sQOyopn1Dg&sa=X&oi=translate&ct=result&resnum=9&ved=0CEQQ7gEwCDgU&prev=/search%3Fq%3DAd%2BTriangulum%26start%3D20%26hl%3Den%26newwindow%3D1%26sa%3DN%26rlz%3D1C1GGGE_enUS400US400%26prmd%3Divb

This web page is #1 in research on medieval free masons geometry

ANTÓNIO ROCHA FADISTA

The geometry of the Duomo was preserved in an edition
of Vitruvius, published in 1521.

Röriczer Matthäus, a Freemason
who has shown his art, breaking his oath of secrecy.Röriczer, who died in 1492, belonged to the third generation
of a family of Master Masons who served in the Cathedral of Regensburg. Although
being a Freemason, and was attached to the oath not to disclose the Masonic
mysteries to the uninitiated, he took a big step with the publication of details
previously hidden in the notebooks of Operative Masonic Lodges

http://translate.google.com/translate?hl=en&sl=pt&u=http://www.maconaria.net/portal/index.php%3Foption%3Dcom_content%26view%3Darticle%26id%3D4&ei=6AnMTOOgIoGusAP-z9CDDw&sa=X&oi=translate&ct=result&resnum=8&ved=0CD4Q7gEwBzge&prev=/search%3Fq%3DAd%2BTriangulum%26start%3D30%26hl%3Den%26newwindow%3D1%26sa%3DN%26rlz%3D1C1GGGE_enUS400US400%26prmd%3Divb

Chapter VI

A NEW ORDER IN SPACE ~

ASPECTS OF A THREEFOLD ORDERING OF

THE FUNDAMENTAL SYMMETRIES OF EMPIRICAL SPACE,

AS EVIDENCED IN THE PLATONIC AND ARCHIMEDEAN POLYHEDRA ~

TOGETHER WITH A TWOFOLD EXTENSION OF THE ORDER
TO INCLUDE THE

REGULAR AND SEMI-REGULAR TILINGS OF THE PLANAR
SURFACE

FIRST PUBLISHED IN THE INTERNATIONAL JOURNAL
OF SPACE STRUCTURES, VOL. 6 NO.1,

UNIVERSITY OF SURREY,

There are generally considered to be five Platonic (Regular) and thirteen
Archimedean (Semi-Regular) polyhedra (leaving aside
the Kepler-Poinsot polyhedra,
and disregarding the regular prisms and antiprisms).

http://web.me.com/rmeurant/INSTITUTE/Natural_Harmony_-_Essaying_Structural_Morphology_files/Ch-06-NewOrderInSpace.pdf

MYSTERIES OF THE EQUILATERAL TRIANGLE

Brian J. McCartin

http://www.m-hikari.com/mccartin-2.pdf

Chartres: the masons
who built a legend, London, by John James 1982,

http://www.johnjames.com.au/currentresearch-art-ge.shtml

Mathes Roriczer Craft Secrets From
Wikipedia

http://en.wikipedia.org/wiki/Mathes_Roriczer

Vitruvii De Architectura

http://www.vitruvius.be/inhboek1.htm

Stereotomy De quinque
corporibus regularibus, De prospectiva pingendi, Vnderweysung der Messung mit den Zirkel un Richtscheyt

Art and science of cutting solids precisely so that their
parts fit together tightly.

http://arts.jrank.org/pages/9896/Stereotomy.html

Piero della Francesca's Tetrahedron Formula

http://www.mathpages.com/home/kmath424.htm

Masion de l’Outil
Tool and Trade Museum

http://www.maison-de-l-outil.com/index.php?page=accueil_anglais

Strictly speaking, the masterwork goes far beyond the accomplishment of a single compagnon. Most of the works that we admire today in museums are very large collective 19th-century works, designed to symbolically mark the supremacy of one guild group over another.

French compagnon master pieces

The first problem that faced medieval builders in the realization
of vaults was how to cut the voussoirs constituting a structure. They seem
to have answered this question from an essentially geometrical point of view,
without taking statical or structural considerations
into account. Indeed, stereotomy treatises illustrate the rules according
to which voussoirs are to be cut in order to solve the different geometrical
problems that may arise.

http://gilbert.aq.upm.es/sedhc/bibli.../CIHC1_017.pdf

A modern exercise in descriptive geometry,
first done by German painter Albrecht Dürer. 21 May 1471 – 6 April 1528)

http://www.rarebookroom.org/Control/duruwm/index.html

The Edinburgh encyclopedia,

British system of projection

Peter Nicholson 1832 -- Freemason

Perspective is only a branch of the doctrine of solids; and
all that this branch teaches, is only the methods for finding the sections
of pyramids and cones, the eye being considered as the vertex, the original
object the base of the pyramid or cone, and the picture to be drawn a section
thereof; the term is there-fore of too general application, perspective being
only a branch of stereography.

The eleventh and twelfth books of the Elements
of Euclid belongs to stereography: these may be looked upon as the theory
of the doctrine of solids, and to them we shall refer our readers for the
original properties; but for their practical applications to useful practical
applications in life, it is rather singular that so little has been done in
this respect. The present article is entirely new. It is of the greatest importance
in the various mechanical departments of architecture. The geometrical principles
in masonry, carpentry, joinery, and the other useful branches of the building
art, are entirely dependent upon it: in short, the cutting of individual pieces
of timber in the art of carpentry, and the formation of separate stones in
masonry, is only the application of stereography to practice.

http://www.archive.org/stream/edinburghencyclo05brew#page/n849/mode/2up

History of geometry

http://en.wikipedia.org/wiki/History_of_geometry

History of Descriptive Geometry in

http://gilbert.aq.upm.es/sedhc/biblioteca_digital/Congresos/CIHC1/CIHC1_121.pdf

A Time Line of Mathematicians

A
Time Line of Mathematicians

LES LIVRES D’ARCHITECTURE

Les Nouvelles Inventions
pour bien bastir…

Philibert DeLorme (c. 1514 – January 8, 1570) was a French architect, one of the great masters of the French Renaissance.

He was born at Lyon, the son of Jean Delorme,
a master mason. At an early age Philibert was sent
to

http://architectura.cesr.univ-tours.fr/traite/Images/Masson643Index.asp

http://sites.google.com/site/delormearchitecture/

Charles-François-Antoine (1780-1854) - Treaty stereotomy : including the applications of descriptive geometry
to the theory of shadows

Traité de stéréotomie : comprenant les applications de
la géométrie descriptive à
la théorie des ombres,
la perspective linéaire, la gnomonique,
la coupe des pierres et la charpente

http://gallica.bnf.fr/ark:/12148/bpt6k109545g.planchecontact.f21.pagination

http://gallica.bnf.fr/RequestDigitalElement?O=NUMM-109545&E=PNG&Deb=21&Fin=21&Param=D

http://gallica.bnf.fr/RequestDigitalElement?O=NUMM-109545&E=PNG&Deb=49&Fin=49&Param=D

Philibert Delorme

http://fr.wikipedia.org/wiki/Philibert_Delorme

Insights and considerations on the "network
key"

Companions stonecutters of ancient Bauhütte

Des Compagnonnages (German)

http://translate.googleusercontent.com/translate_c?hl=en&sl=fr&u=http://www.compagnonnage.info/compagnons-tailleurs-de-pierre/reseaux.htm&prev=/search%3Fq%3DAd%2BQuadratum,%2BAd%2BTriangulum%26start%3D20%26hl%3Den%26newwindow%3D1%26sa%3DN%26prmd%3Divb&rurl=translate.google.com&twu=1&usg=ALkJrhhKhF_Brtoas7hIi0Kp59uQQvXqrA

Wooden churches in

http://commons.wikimedia.org/wiki/Category:Wooden_churches_in_Romania

Tracing a Sacred Building Tradition

http://www.lub.lu.se/luft/diss/tec_797/tec_797.pdf

Note:
the above link is 140MB... , but really neat timber
framing in

De Symmetria.
. . and Underweysung der
Messung- 1538 - Dürer, Albrecht
(author) - Nuremberg - The Warnock Library

Rare Books

Page 113 seed of life, daisy wheel or just Euclidean geometry

http://www.rarebookroom.org/Control/duruwm/index.html

Elementa Geometriae-
1482 - Euclid, (author) - Venice - The Bancroft Library; University of California,
Berkeley

Rare Books

http://www.rarebookroom.org/Control/eucgeo/index.html

Descriptive geometry, as applied to the drawing
of fortification and sterotomy. For the use of the
cadets of the U.S. Military academy (1864)

Mahan, D. H. (Dennis Hart), 1802-1871

http://www.archive.org/stream/descriptivegeom02mahagoog#page/n75/mode/1up

BOOKS ON ARCHITECTURE

http://architectura.cesr.univ-tours.fr/traite/matiere.asp

Derand, Francis (1588?
-1644)

The architecture of arches, or the art features and cutting vaults ..., Par
is, Sebastien Cramoisy, 1643.

http://architectura.cesr.univ-tours.fr/traite/Notice/B250566101_11598.asp?param=

Creating Shapes in Civil and Naval Architecture:
A Cross-Disciplinary Comparison

By Horst Nowacki, Wolfgang Lefèvre

Page 302 http://books.google.com/books?id=8FoHYXEwAXEC&lpg=PP1&pg=PA108#v=onepage&q&f=false

Stereotomy, construction,stone-cutting(1567-2002)

http://www.arch.unige.it/bma/ENGL/engl_biblio_stereo_crono.html

The trunk of the street in Paris Larivière

http://translate.google.com/translate?hl=en&sl=fr&u=http://compagnonnage.info/blog/blogs/blog1.php/2009/10/05/trompe-rue-lariviere-paris&ei=E1TLTP6XKYe8sAOEsOTQDg&sa=X&oi=translate&ct=result&resnum=2&sqi=2&ved=0CB8Q7gEwAQ&prev=/search%3Fq%3Dtrompe%2Bde%2Bla%2Brue%26hl%3Den%26newwindow%3D1%26rlz%3D1C1GGGE_enUS400US400%26prmd%3Div

The Construction of Gothic Cathedrals byJohn Fitchen

http://books.google.com/books?id=KI3dJzP0p00C&printsec=frontcover&dq=The+Construction+of+Gothic+Cathedrals+by+John+Fitchen&source=bl&ots=5QbsncAK7E&sig=kObljaXP_pDy2l5MDh5fwWIV3dk&hl=en&ei=aDLMTOW_N5KCsQO2tezQDg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBsQ6AEwAA#v=onepage&q&f=false

The New Metal Worker Pattern Book | by George Watson Kittredge 1901

Problem 146. Pattern Of A Tapering; Article With
Equal Flare Throughout, Which Corresponds To The Frustum Of A Cone Whose Base
Is An Approximate Ellipse Struck From Centers, The Upper Plane Of The Frustum
Being Oblique To The Axis

http://chestofbooks.com/crafts/metal/Metal-Pattern/Problem-146-Pattern-Of-A-Tapering-Article-With-Equal-Flare-Throughout-Which-C.html

Apollonius
on conic sections from ~200 BC

http://www.archive.org/stream/treatiseonconics00apolrich#page/n9/mode/2up

Villard deHonnecourt’s notebooks.

Gaspard Monge, no Freemason tattler, who wrote about "Descriptive Geometry,
or the Art & Sciece of Masonic Symbolism."

Carpentry and building, Volume 13 1891

MASONRY AND STONE CUTTING

http://books.google.com/books?id=AXoAAAAAYAAJ&lpg=PA142&ots=78BexfwVNM&dq=conoid%20%20vault&pg=PA111#v=onepage&q=conoid%20%20vault&f=false

Călineşti Susani church sun rosette or Thunder rosette

http://en.wikipedia.org/wiki/C%C4%83line%C5%9Fti_Susani_church

Reconstructing Architectural Geometry

Earl Mark, Ph.D.

Associate Professor of Architecture

University of Virginia

http://www.re-h.nl/koutamanis/image05/mark/arqpapCMP.pdf

Wood carving in churches in

http://commons.wikimedia.org/wiki/Category:Wood_carving_in_churches_in_Romania

Gothic Cathedral and Church Construction

http://www.abelard.org/france/cathedral-construction.php

Schneider, Mark. "Self-Invention and Deviance:
Philibert de l'Orme's Role in the
Creation of the Savant Professional Architect."Discoveries25.1
(2008). 24 June 2008.

*Le Premier Tome de l’architecture*

trompes(ribless, conical vaults)

http://cstl-cla.semo.edu/reinheimer/discoveries/archives/251/schneider251pf.htm

Abstract.Roger Herz-Fischler
presents a revised version of a chapter entitled "Proportions" that
appeared in the problems part of his book,Space, Shape and Form /An Algorithmic
Approach, developed for a mathematics course he taught in the School of Architecture
at Carleton University from 1973-1984.

http://www.emis.de/journals/NNJ/Didactics-RHF.html

The Role of the Master Mason in Mediaeval English
Building

L. R. Shelby

Vol. 39, No. 3 (Jul., 1964), pp. 387-403

(article consists of 17 pages)

Published by:Medieval
Academy of America

Stable URL: http://www.jstor.org/stable/2852495

Ad Quadratum Construction and Study of the Regular Polyhedra by Jean Le Mée

http://www.gatewaycoalition.org/files/millennium_sphere/products/AdQuadratum.pdf

TECHNICAL DRAWING (2) The first half of the 19th
century might well be called a formative period in the development of technical
drawing. Durer (1471-1528) was credited with the
first basic knowledge of orthogonal/orthographic projection.

But nothing was published until 1795 when a book written by Gaspard Monge,
'La Geometries Descriptive'.

http://www.gr8lessons.com/history/TechDraw2.html

A history of Western architectureBy David Watkin Kuttenberg Cathedral

A history of architecture in
all countries from the earliest times ..., Volume 2

By James Fergusson Saint Barbara Kuttenberg

http://books.google.com/books?id=ENgDAAAAYAAJ&lpg=PA74&ots=aZG1AjCAGS&dq=Saint%20Barbara%20Kuttenberg&pg=PA136#v=onepage&q&f=false

Centralblatt
der Bauverwaltung, Volume
8 Saint Barbara Kuttenberg

http://books.google.com/books?id=PJkAAAAAMAAJ&dq=Saint%20Barbara%20Kuttenberg&pg=PA100#v=onepage&q&f=false

Marcus Vitruvius Pollio:de
Architectura, Book I

ichnography, orthography, and scenography

Arises out of dimension(quantitas)

The form of a theater is to be adjusted so that from the center of the dimensionless
allotted to the base of the perimeter is a circle to be described, in Which
are inscribed four equilateral triangles, at equal distances from each other,
Whose points are to touch the circumference of the circle.This is the method
that is practiced by astrologers in describing the twelve celestial signs,
according to the musical division of the constellations.Of these triangles,
the side of That which is nearest the scene will
Determine the face thereof in that part where it cuts the circumference of
the circle.Then the center is a line drawn through parallel to it, Which Will separate thepulpitumof
theprosceniumfrom theorchestra.

http://translate.googleusercontent.com/translate_c?hl=en&sl=de&u=http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Vitruvius/1*.html&prev=/search%3Fq%3Dalso%2BMarcus%2BVitruvius%2BPollio,%2BVitruvii%2BDe%2BArchitectura%2B-chpt%2B8%26hl%3Den%26newwindow%3D1%26prmd%3Divb&rurl=translate.google.com&twu=1&usg=ALkJrhicxjKBP8UiP-k09rXBkCx0-z43-A

Cistercian benedictine
monks

Constantine the Great

Hagia Sophia

Byzantine architecture

Constantinople

Basilica Cistern 336 columns in 12 rows,

http://www.arkeo3d.com/byzantium1200/introduction.html

The intersection of cylinders (i.e. a groin vault) was first analysed
precisely byPiero della
Francesca(c.1415–92) in his worked example for Problem X inDe quinque
corporibus regularibus, and their perspectival
representation inDe prospectiva pingendi(II and XI).Albrecht Dürer(1471–1528) introduced Euclidean geometry to northern
artists inVnderweysung der Messung mit
den Zirkel un Richtscheyt(1525),
which became more accessible outside Germany in a Latin translation,Institutiones Geometricae(Paris,1538).

Abstract.Answers to a reader's query about the origin and
symbolic significance of pointed arches, in the Nexus Network Journal.

http://www.emis.de/journals/NNJ/Query01-PointedArches.html

Euclid’s Elements Book 1 Prop
1

*To construct an equilateral triangle on a given finite
straight line.*

http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI1.html

Knights Templar Sinclair

http://www.rosslyntemplars.org.uk/index.php/sinclairs/

The construction of Gothic cathedrals:
a study of medieval vault erection

By John Fitchen

http://books.google.com/books?id=KI3dJzP0p00C&lpg=PA42&ots=5QbrpdEF8C&dq=medieval%20cathedral%20vaulting&pg=PA57#v=onepage&q=medieval%20cathedral%20vaulting&f=false

Tiles as a substitute for steel: the art of the timbrel vault

http://www.lowtechmagazine.com/2008/11/tiles-vaults.html

Perspective versus Stereotomy:
From Quattrocento Polyhedral Rings to Sixteenth-Century
Spanish Torus Vaults José Calvo-López

http://www.springerlink.com/content/3347m764427x4230/fulltext.pdf

Dürer himself called the ellipse “the egg line” (eyer lini).

http://pavlopoulos.wordpress.com/2009/07/27/albrecht-duerers-eyer-lini/

Guldin in 1640 who discovered the elliptical
nature of the curve

A treatise on the teeth of wheels:Demonstrating the best forms which
can be given to them for the purposes of machinery, such as mill-work and
clock-work

http://books.google.com/books?id=-5FKAAAAMAAJ&lpg=PA144&ots=crRqxNmgWQ&dq=treatise%20of%20geometry%20Charles-%C3%89tienne-Louis%20Camus&pg=PA121#v=onepage&q&f=false

Marcus Vitruvius Pollio:

De Architectura, Liber III

Chapter 1

3rd

Just so the parts of Temples should correspond with each other, and with the
whole. The navel is naturally placed in the center of the human body, and,
if in a man lying with his face upward, and his hands and feet extended, from
his navel as the center, a circle be described, it will touch his fingers
and toes. It is not alone by a circle, that the human body is circumscribed
Thus, as may be seen by placing it within a square. For measuring from the
feet to the crown of the head, and then across the arms fully extended, we
find The latter measure equal to the former, so that lines at right angles
to each other, enclosing the figure, will form a square. º

vero similiter sacrarum aedium membra ad Universam summam ex partibus singulis
totius magnitudinis convenientissimum debent habere commensus responsum. item
centrum corporis medium naturaliter est umbilicus. inamque si fuerit homo
conlocatus supinus manibus et Pedibus pansis circinique conlocatum centrum
eius in umbilico, circumagendo rotundationem utrarumque manuum et digiti pedis
linea tangentur. non minus quemadmodum schematic rotundationis in corpore
efficitur, item quadrata designatio in eo invenietur. nam si ad summum caput
a Pedibus imis mensum erit eaque mensura relata ad manus fuerit Panza, invenietur
latitudo uti eadem altitudo, quemadmodum Areae quae sunt ad normam quadrata.

Vitruvius
Book 6

Properties of two famous amphitheatres : the Colosseum(Rome) and the Arena(Verona)

Elliptical Colosseum

The Sambuca is a elliptical
curve with ordinates or a triangular harp

Sambuca

gnomon,menaeus,ecliptic, analemmas,
elliptical trace on an analemmatic sundial,16th century works of Oronce Fine,
shadow length of the gnomon, the hectemoros circle

Vitruvius' analemma, Ptolemy's anaemma, water clocks (clepsydras)

The menaeus rotated edge-on is shown, then two edge-on and one facing ecliptic
circles oriented at different angles relative to the menaeus but having the
same diameter projection. We can drop perpendiculars to the diameter LH to
find the sun circles

The
Analemmas of Vitruvius and Ptolemy

Vitruvius Pollio and the Analemma

orthographic projection

The graphical construction called the épure in France for the design of sundials
is a similar procedure.

The analemma was also a kind of armillary sphere used for spherical trigonometry
by Ptolemy

*Roman engineers used a method of graphical computation and design that
was very much in the style of Monge's descriptive geometry of the 19th century,
until recently taught to every engineering student. This procedure makes use
of two or more views, each showing two dimensions, and the combination showing
all three dimensions necessary to describe a body in space. Vitruvius would
probably have made his views overlap, in spite of the confusion of lines that
results, to save space. We shall separate the two views in the modern manner
so that each can be appreciated alone. The graphical construction called the
épure in France for the design of sundials is a similar procedure. *

Vitruvius Pollio and the Analemma

The orthographic projection was known by the Egyptians and Greeks 2000 years ago. Azimuthal Projections

41 BC -- The Roman architect Vitruvius (Marcus Vitruvius Pollio) includes information on sundials in his "De Architectura" on engineering and architecture.
He uses the word analemma in referring to a form of orthographic projection.

orthography: the art of drawing anything without perspective, as though viewed from infinity. In dialling, the sphere so drawn consists of circles, straight lines and ellipses. Hence orthographic (or orthogonal) projection, which is used in the universal astrolabe.

analemmatic dials consisting of hour points, (rather than lines) laid round an ellipse

Regiomontanus, Apian and Capuchin Sundials

Regiomontanus, Apian and Capuchin Sundials

Capuchin Friar

Analemmatic sundial by plotting abscissas and ordinates. Analemmatic Dial. Motto: The Light and Shadow of God

Analemmatic sundials: How to build one and why they work

The analemmatic sundial is based on an ellipse which is marked out in hour points

An ellipse is an example of a curve called a conic section. Other conic sections are the circle, the parabola and the hyperbola. The Earth travels around the sun on an elliptical path.

Analemmatic sundials

umbra recta: Latin for "upright shadow", it is the label often found on the cotangent scale of altitudes < 45º on a shadow square.

umbra versa: Latin for "reverse shadow", it is the label often found on the tangent scale of altitudes > 45º on a shadow square.

Basic sundial nomenclature

umbra recta -- the shadow cast by a horizontal stick on a vertical surface

umbra versa -- the shadow cast by a vertical stick on a horizontal surface

Quadrato geometrico per misurare ogni lunghezza (geometrical square to measure every length) 60

Scala altimetra di gradi 100 per misurare ogni altezza da lontano (Altitude scale of 100 degrees, to measure every far height) 12 .

The Shadow Square

The shadow square, also called altitude scale, generally placed on the back
of astrolabes and quadrants, is the basic element of the so-called geometrical
square. The instrument serves to measure heights and distances by simulating
the ratio between a gnomon and its shadow. The umbra recta simulates the shadow
cast on the horizontal plane by a vertical gnomon when the Sun's ray is inclined
between 0° and 45°. The umbra versa simulates the shadow cast on the vertical
plane by a horizontal gnomon when the Sun's ray is inclined between 45° and
90°. To each value of the umbra recta corresponds a value of the umbra versa.
When the ray is inclined by 45°, the two shadows are equal (umbra media).

"Ecliptic" simply means "(oblique) circle" (in Greek ekklitiké, from klinein = to bend)

Claudius Ptolemaeus (Ptolemy) epicycles

Claudius Ptolemaeus (Ptolemy) Apotelesmatika -Quadripartitum (four books) of astrological treatise

CHRONOLOGY – some selected dates in the development of sundials and solar astronomy

development of sundials and solar astronomy

Analemmatic Dial. Motto: The Light and Shadow of God

Analemmatic sundial by plotting abscissas and ordinates. The point (3,2) has 3 as its abscissa (x) and 2 as its ordinate (y).

Drive the chariot of the sun’s arc south of the equator to see the chariot of light at winter solstice.

Azimuthal Projections

Equatorial orthographic map, central meridian 70°E It is remarkable that some azimuthal projections, important even today, were known more than two thousand years ago by the Greeks and maybe by the Egyptians. The orthographic, stereographic and gnomonic projections are all based on solid principles of perspective and Euclidean geometry, while the azimuthal equidistant dates from the 15th century and is constructed arbitrarily.

Mentioned by the Greek Hipparchus in the 2nd century B.C., but probably known earlier, the azimuthal orthographic (usually simply referred to as the orthographic) projection was called analemma by Ptolemy and got its modern name from d'Aiguillon (1613).

Map Projections

mubattakh == melon shape, flatten mubattakh -- ellipse orthographic projection

reconstruction of Al-Birunis Cylindrical Projection -- 442 AD

Islam Anaphoric clock dials -- Astrolabe

Cartography in the traditional Islamic , map making

The history of cartography: Cartography in the traditional ..., Volume 2, Book 1 edited by J. B. Harley, David Woodward

AXONOMETRIC PROJECTION

The Greek word axon means axis and metric means to measure. Axonometric projection is
a parallel projection technique used to create pictorial drawings of objects by rotating the
object on an axis relative to a projection plane to create a pictorial view.

Axonometric drawings are classified by the angles between the lines comprising the
axonometric axes.

AXONOMETRIC PROJECTION

Taprats is a Java applet that implements one such design technique for Islamic
star patterns for polygons tiling.

Taprats Computer
Generated Islamic Star Patterns

Islamic star patterns Java Applet

six-point geometry, eight-point geometry, ten-point geometry, twelve-point
rosettes,fifteen-point rosette, Islamic geometric design,

Distinct from sacred geometries is geomancy, a tradition of divination, but
which has a tradition in the Arabic and other worlds with a relationship with
numbers, not geometry. The divination is composed of two elements: numbers
and a body of knowledge governing interpretation. The only reason I mention
it here is that some believe there is a relationship between geomancy and
mathematics and, by extension, astrology and cosmology to which sacred geometries,
as I’ve mentioned, relate.

Here the same basic construction has been developed in order to produce a
shape with six pointed foils – a hexafoil or sexfoil.

Arabic geometry,the
basic seven-circle rose from the six-point geometry

based on eleven-point geometry. The hendecagon, or eleven-sided figure, has
internal angles of 147·2727…°.

Hexagram

mandala , [Skt.,=circular, round] a concentric diagram , Sacred Islamic Geometry grows from a circle

star of David or Solomon's seal

The works of Euclid and Pythagoras were among the first to be translated into Arabic.

The rules of construction of geometric patterns provide a visual analogy to religious rules of behavior

The circle, and its centre, are the point at which all Islamic patterns begin.

From the circle comes three fundamental figures in Islamic art, the triangle, square and hexagon. The triangle by tradition is symbolic of human consciousness and the principle of harmony. The square, the symbol of physical experience and the physical world-or materiality-and the hexagon, of Heaven. Another symbol prevalent in Islamic art is the star and has been the chosen motif for many Islamic decorations. In Islamic iconography the star is a regular geometric shape that symbolizes equal radiation in all directions from a central point. All regular stars -- whether they have 6, 8, 10, 12, or 16 points -- are created by a division of a circle into equal parts. The center of the star is center of the circle from which it came, and its points touch the circumference of the circle. The rays of a star reach out in all directions, making the star a fitting symbol for the spread of Islam.

ISLAMIC PATTERNS & GEOMETRY

**Railroads and Skew Vaults **

The difficulty of constructing skewed vaults led to interesting solutions in Gothic architecture, as John Fitchen shows in his book The Construction of Gothic Cathedrals: A Study in Medieval Vault Erection. Similar problems arose in the nineteenth century, when “the advent of railroads demanded bridges of arched masonry many of which had to accommodate a right-of-way that crossed a stream or a roadway at an oblique angle.”

The shapes of the groin voussoirs, if properly cut to key accurately into
the web stones of the two vault surfaces whose intersection they form, are
very difficult to arrive at. This difficulty unquestionably accounts for the
very small number of unribbed, simple groin vaults constructed by the medieval
builders over naves. **They could “get away with” approximate shapes
**

Islamic Art and the Argument from Academic Geometry

10th century treatise "On Those Parts of Geometry Needed by Craftsmen" by the famous Persian mathematician Abul Wafa al-Buzjani at Baghdad

Islamic Constructions: The Geometry Needed by Craftsmen

Geometric Constructions and their Arts in Historical Perspective

The Circle for Iranian Architectural Studies

Spherical tetrahedron, Tiling with six spherical squares, Construction of Some Spherical Archimedean Solids

An Introduction to Medieval Spherical Geometry for Artists and Artisans

The apothem of a regular polygon is the distance from center to the midpoint of a side. It is also the radius of the inscribed circle. Only regular polygons have an apothem.

Platonic solids

Another important work in Islamic geometry was Apollonios of Perga's The Conics from about 200 BC. Though The Conics contained eight chapters or books, only four exist in Greek and only seven in Arabic. These three Greek scholars -- Euclid, Archimedes, and Apollonios -- formed the basis of Islamic mathematics. Muslim mathematicians and translators are responsible for the preservation and transmission of these texts through the medieval period.

Islamic geometry History

Ibrahim ibn Sinan ibn Thabit ibn Qurra (d. 946), On Drawing the Three Conic Sections

When a right triangle is rotated about the X-axis, it forms a cone with its axis of symmetry as the same X-axis.

three centered elliptical, five centred elliptical arch

Definition of the Ellipse

The important thing to remember is that conic sections in Roman times were all defined as cuts of cones; equation forms did not exist! So laying out an elliptical field is something that may not have even entered someone's mind.

Stereographic projection

Earliest known uses were in Greece for map making
Earliest references in literature (Roman, ~100 B.C.)
Vitruvius ~ Ten Books on Architecture
Ptolemy’s ~ Representation of the Sphere in the plane

Building
age, Volume 10 -- 1888

What is this geometrical symbol in Roncesvalles?

2* pi *height = perimeter of pyramid

1/2 * base * apothem = phi

The
Great Pyramid, The Great Discovery, and The Great Coincidence

The Great and second Great Pyramid Geometry

Knights Templar tombstone from St Magnus cathedral in Kirkwall

1118 Knights Templar Founded in Jerusalem, at the site of Solomon's Temple.

845 Saint Clair Eventual namesake of the Sinclair family is born.

Knights Templar tombstone from St Magnus cathedral in Kirkwall

History of Building construction 60,000 BC to 1000 BC
History of Building construction

History of Building Greece Log Motif

The Way of the Japanese Carpenter: Tools and Japanese Architecture. by William
H. Coaldrake

Chris
Hall Tressle Saw Horse

Traite Theorique Et Pratique De Charpente by Louis Mazerolle(1889) and Art
Du Trait Pratique De Charpente by Emile Delataille

Chris
Hall Emile Delataille

Who is Emile DELATAILLE

Title : Traité de stéréotomie : comprenant les applications
de la géométrie descriptive à la théorie des ombres,
la perspective linéaire, la gnomonique, la coupe des pierres et la
charpente, avec un atlas composé de 74 planches in-folio. II. Atlas.
- In-fol., 74 pl. / par C.-F.-A. Leroy,...

Author : Leroy, Charles-François-Antoine (1780-1854)

Publisher : Gauthier-Villars (Paris)

Date of publication : 1877

Contributor : Martelet, Émile (1805-18..). Éditeur scientifique.
Annotateur

Subject : Géométrie descriptive

Subject : Stéréotomie

Type : monographie imprimée

Language : French

Format : 2 vol. : fig. ; in-4, in-fol.

Format : application/pdf

Copyright : domaine public

Traité de stéréotomie for charpente

Good book on stereotomic geometry by Leroy, Charles-François-Antoine

Traité de géométrie descriptive

Traité
de géographie descriptive By Jules De La Gournerie

a corona ("crown") and cyma ("ogee") molding to support
the projecting roof.

2. For the human body is so designed by nature that the face, from the chin to the top of the forehead and the lowest roots of the hair, is a tenth part of the whole height; the open hand from the wrist to the tip of the middle finger is just the same; the head from the chin to the crown is an eighth, and with the neck and shoulder from the top of the breast to the lowest roots of the hair is a sixth; from the middle of the breast to the summit of the crown is a fourth. If we take the height of the face itself, the distance from the bottom of the chin to the under side of the nostrils is one third of it; the nose from the under side of the nostrils to a line between the eyebrows is the same; from there to the lowest roots of the hair is also a third, comprising the forehead. The length of the foot is one sixth of the height of the body; of the forearm, one fourth; and the breadth of the breast is also one fourth. The other members, too, have their own symmetrical proportions, and it was by employing them that the famous painters and sculptors of antiquity attained to great and endless renown.

http://www.crystalinks.com/greekarchitecture.html

13. All members over the capitals of columns, such as architraves, friezes,
coronæ, tympana, crowning members (fastigia), and acroteria, should not
be vertical, but** inclined forwards**, each a** twelfth
part of its height**; and for this reason, that when two lines are produced
from the eye, one to the upper part of a member, and the other to its lower
part, the upper line or visual ray will be longer than the lower one, and if
really vertical, the member will appear to lean backwards; but if the members
are set out as above directed, they will have the appearance of being perpendicular.

13. Membra omnia quae supra capitula columnarum sunt futura id est epistylia zophoroe coronae tympana fastigia acroteria inclinanda sunt in fronte suae cuiusque altitudinis parte XII, ideo quod cum steterimus contra frontes, ab oculo lineae duae si extensae fuerint et una tetigerit imam operis partem, altera summam, quae summam tetigerit longior fiet. ita quo longior visus linea in superiorem partem procedit, resupinatam facit eius speciem. cum autem, uti supra scriptum est, in fronte inclinata fuerint, tunc in aspectu videbuntur esse ad perpendiculum et normam.

1/7 = 52°

1/9 = 38°

http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Vitruvius/3*.html

The famous controversial conferences at Milan cathedral council between 1389
- 1401

Picturing
machines 1400-1700 By Wolfgang Lefèvre

Just ordered Picturing machines 1400-1700 By Wolfgang Lefèvre

MIT Press, 2004 - Computers - 347 pages

Technical drawings by the architects and engineers of the Renaissance made use
of a range of new methods of graphic representation. These drawings—among them
Leonardo da Vinci's famous drawings of mechanical devices—have long been studied
for their aesthetic qualities and technological ingenuity, but their significance
for the architects and engineers themselves is seldom considered. The essays
in Picturing Machines 1400-1700 take this alternate perspective and look at
how drawing shaped the practice of early modern engineering. They do so through
detailed investigations of specific images, looking at over 100 that range from
sketches to perspective views to thoroughly constructed projections.

In early modern engineering practice, drawings were not merely visualizations of ideas but acted as models that shaped ideas. Picturing Machines establishes basic categories for the origins, purposes, functions, and contexts of early modern engineering illustrations, then treats a series of topics that not only focus on the way drawings became an indispensable means of engineering but also reflect the main stages in their historical development. The authors examine the social interaction conveyed by early machine images and their function as communication between practitioners; the knowledge either conveyed or presupposed by technical drawings, as seen in those of Giorgio Martini and Leonardo; drawings that required familiarity with geometry or geometric optics, including the development of architectural plans; and technical illustrations that bridged the gap between practical and theoretical mechanics.

The American house-carpenter: a treatise upon architecture, cornices and mouldings, framing, doors, windows, and stairs : together with the most important principles of practical geometry ... By Robert Griffith Hatfield 1845

carpentry double curvature arch

Tudor Arch

Ogee Arch

Rampant Arch

Lancet Arch

Gothic Arch

Morrish Arch

Semicircle Arch

Segment Arch

Pointed Arch

Elliptical Arch

Equilateral Arch

Double Curvature Arch

Trompe Arch

Catenary Arch

http://books.google.com/books?id=LqIJAAAAIAAJ&dq=related%3AISBN0748765328&pg=PA69#v=onepage&q&f=false

57.—The axis of a cone is a right line passing through it, from the vertex
to the centre of the circle at the base.

58.—An ellipsis is described if a cone be cut by a plane, not parallel to its
base, passing quite through the curved surface. (a b, Fig. 23.)

59.—A parabola is described if a cone be cut by a plane, parallel to a plane
touching the curved surface. (c d, Fig. 23— c d being parallel to fg.)

60.—An hyperbola is described if a cone be cut by a plane, parallel to any plane
within the cone that passes through its vertex. (e h, Fig. 23.)

61.—Foci are the points at which the pins are placed in describing an ellipse.
(See Art. 115, and/,/, Fig. 24.)

62.—The transverse axis is the longest diameter of the ellipsis. (a b, Fig.
24.)

63.—The conjugate axis is the shortest diameter of the ellipsis ; and is, therefore,
at right angles to the transverse axis. (c d, Fig. 24.)

64.—The parameter is a right line passing through the focus of an ellipsis,
at right angles to the transverse axis, and terminated by the curve. (g h and
g t, Fig. 24.)

65.—A diameter of an ellipsis is any right line passing through the centre,
and terminated by the curve. (k I, or m n, Fig. 24.)

66.—A diameter is conjugate to another when it is parallel to a tangent drawn
at the extremity of that other-—thus, the diameter, m n, {Fig. 24,) being parallel
to the tangent, o p, is therefore conjugate to the diameter, k I.

67.—A double ordinate is any right line, crossing a diameter of an ellipsis,
and drawn parallel to a tangent at the extremity of that diameter. (i t, Fig.
24.)

68.—A cylinder is a solid generated by the revolution of a right-angled parallelogram,
or rectang.e, about one of its sides; and consequently the ends of the :ylinder
are equal circles. [Fig. 25.)

69.—The axis of a cylinder is a right line passing through it, from the centres
of the two circles which form the ends.

70.—A segment of a cylinder is comprehended under three planes, and the curved
surface of the cylinder. Two of these are segments of circles : the other plane
is a parallelogram, called by way of distinction, the plane of the segment.
The circular segments are called, the ends of the cylinder. {Fig. 26.)

Architectural notes on German churches and vaults By William Whewell 1835 Architectural notes on German churches