21. Ceva_Giovanni Biography of giovanni ceva (16471734) giovanni ceva was educated in a Jesuitcollege in Milan, then studied at the university of Pisa. http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Ceva_Giovanni.html

Giovanni Ceva Born: 7 Dec 1647 in Milan, Italy Died: 15 June 1734 in Mantua, Italy Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index Version for printing Giovanni Ceva was educated in a Jesuit college in Milan, then studied at the university of Pisa. He taught at Pisa before being appointed Professor of mathematics at the University of Mantua in 1686, a post he held for the rest of his life. When appointed in 1686 Giovanni Ceva served the Gonzagas rulers. However in 1708 Austria annexed the duchy and began to construct heavy fortifications. Giovanni Ceva quickly moved to support the new Austrian regime. For most of his life Giovanni Ceva worked on geometry. He discovered one of the most important results on the synthetic geometry of the triangle between Greek times and the 19 th Century. The theorem states that lines from the vertices of a triangle to the opposite sides are concurrent precisely when the product of the ratio the sides are divided is 1. He published this in De lineis rectis Ceva also rediscovered and published Menelaus 's theorem. He also studied applications of mechanics and statics to geometric systems. Although he wrongly concluded that the periods of oscillation of two pendulums were in the same ratio as their lengths, he later corrected the error.

22. References For Ceva_Giovanni References for the biography of giovanni ceva. G Loria, Per la biografia degiovanni ceva, Rendiconti dell istituto lombardo di scienze e lettere 48 http://www-groups.dcs.st-and.ac.uk/~history/References/Ceva_Giovanni.html

References for Giovanni Ceva Version for printing Biography in Dictionary of Scientific Biography (New York 1970-1990). Articles: A Brigaglia and P Nastasi, The solutions of Girolamo Saccheri and Giovanni Ceva to Ruggero Ventimiglia's 'Geometram quaero' : Italian projective geometry in the late seventeenth century (Italian), Arch. Hist. Exact Sci. G Loria, Per la biografia de Giovanni Ceva, Rendiconti dell'istituto lombardo di scienze e lettere A Procissi, Di alcune lettere di Giovanni Ceva, Atti Secondo Congresso Un. Mat. Ital. (Rome, 1942), 895-896. A Procissi, Lettere di Giovanni Ceva ad A Magliabechi con note bibliografiche, Period. Mat. Main index Birthplace Maps Biographies Index History Topics ... Anniversaries for the year JOC/EFR December 1996 School of Mathematics and Statistics University of St Andrews, Scotland The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/References/Ceva_Giovanni.html

23. Allmath.com - Math Site For Kids! Home Of Flashcards, Math ceva, giovanni. chayva. (?16471734). Geometer, born in Milan, Italy. He gavehis name to a theorem on concurrent lines through the vertices of a http://www.allmath.com/biosearch.php?QMeth=ID&ID=6440

24. Biografia De Ceva, Giovanni Translate this page ceva, giovanni. (Milán, 1648-Mantua, 1734) Matemático italiano. Profesor dematemática y también ingeniero dedicado a la construcción de obras hidráulicas, http://www.biografiasyvidas.com/biografia/c/ceva.htm

25. Ceva, Giovanni --  Encyclopædia Britannica ceva, giovanni Italian mathematician, physicist, and hydraulic engineer bestknown for the geometric theorem bearing his name concerning straight lines that http://www.britannica.com/eb/article-9002192

Home Browse Newsletters Store ... Subscribe Already a member? Log in Content Related to this Topic This Article's Table of Contents Giovanni Ceva Print this Table of Contents Shopping Price: USD $1495 Revised, updated, and still unrivaled. The Official Scrabble Players Dictionary (Hardcover) Price: USD $15.95 The Scrabble player's bible on sale! Save 30%. Merriam-Webster's Collegiate Dictionary Price: USD $19.95 Save big on America's best-selling dictionary. Discounted 38%! More Britannica products Ceva, Giovanni  Encyclopædia Britannica Article Page 1 of 1 Giovanni Ceva born 1647/1648, Milan [Italy] died 1734, Mantua [Italy] Italian mathematician, physicist, and hydraulic engineer best known for the geometric theorem bearing his name concerning straight lines that intersect at a common point when drawn through the vertices of a triangle. Ceva, Giovanni... (75 of 353 words) var mm = [["Jan.","January"],["Feb.","February"],["Mar.","March"],["Apr.","April"],["May","May"],["June","June"],["July","July"],["Aug.","August"],["Sept.","September"],["Oct.","October"],["Nov.","November"],["Dec.","December"]];

26. Croce, Giovanni --  Encyclopædia Britannica Croce, giovanni composer who, with Andrea and giovanni Gabrieli, was one of theleading Venetian giovanni ceva University of St Andrews, Scotland http://www.britannica.com/eb/article-9027940

Home Browse Newsletters Store ... Subscribe Already a member? Log in Content Related to this Topic This Article's Table of Contents Giovanni Croce Print this Table of Contents Shopping Price: USD $1495 Revised, updated, and still unrivaled. The Official Scrabble Players Dictionary (Hardcover) Price: USD $15.95 The Scrabble player's bible on sale! Save 30%. Merriam-Webster's Collegiate Dictionary Price: USD $19.95 Save big on America's best-selling dictionary. Discounted 38%! More Britannica products Croce, Giovanni  Encyclopædia Britannica Article Page 1 of 1 Giovanni Croce born c. 1557, Chioggia, near Venice [Italy] died May 15, 1609, Venice also called Chiozzotto composer who, with Andrea and Giovanni Gabrieli, was one of the leading Venetian composers of his day. Croce, Giovanni... (75 of 119 words) var mm = [["Jan.","January"],["Feb.","February"],["Mar.","March"],["Apr.","April"],["May","May"],["June","June"],["July","July"],["Aug.","August"],["Sept.","September"],["Oct.","October"],["Nov.","November"],["Dec.","December"]]; To cite this page: MLA style: "Croce, Giovanni."

27. Ceva, Giovanni, Geometria Motus, 1692 ceva, giovanni, Geometria motus, 1692. Placed in the Public Domain by Max PlanckInstitute for the History of Science http://echo.mpiwg-berlin.mpg.de/content/historymechanics/archimdesecho/cevag_geo

Ceva, Giovanni Geometria motus Placed in the Public Domain by Max Planck Institute for the History of Science get full text XML This resource is part of The Collection Browser of the Archimedes Project Max Planck Institute for the History of Science The Archimedes Project is the digital library component of a major research project of the Max Planck Institute for the History of Science dealing with mental models in the history of mechanics. The library contains key primary sources documenting the development from ancient to early modern mechanics. The collection browser of the project coordinates images and text and links the texts by means of language technology to dictionaries.

28. The Collection Browser Of The Archimedes Project ceva, giovanni, Geometria motus, 1692. get full text XML Placed in the PublicDomain by Max Planck Institute for the History of Science http://echo.mpiwg-berlin.mpg.de/content/historymechanics/archimdesecho

ECHO Content Graphical Overview ECHO Technology Graphical Overview ... Agricola, De re metallica Agricola, Georgius De re metallica get full text XML Placed in the Public Domain by Max Planck Institute for the History of Science Agricola, De re metallica (engl.) Agricola, Georgius De re metallica get full text XML Placed in the Public Domain by Max Planck Institute for the History of Science Alberti, De re aedificatoria Alberti, Leone Battista De re aedificatoria get full text XML Placed in the Public Domain by Max Planck Institute for the History of Science Alberti, De re aedificatoria (engl.) Alberti, Leone Battista Architecture Placed in the Public Domain by Max Planck Institute for the History of Science Babington, Pyrotechnia Babington, John Pyrotechnia Placed in the Public Domain by Max Planck Institute for the History of Science Baif, De re navali commentarius Baif, Lazare de De re navali commentarius get full text XML Placed in the Public Domain by Max Planck Institute for the History of Science Baldi, Mechanica Aristotelis Baldi, Bernardino In mechanica Aristotelis problemata exercitationes get full text XML Placed in the Public Domain by Max Planck Institute for the History of Science Baliani, De motu (1638)

29. INTRODUZIONE1 giovanni Olivero, Bonifacio Giuniore, Oberto, Oddone ed Enrico dei marchesi di ceva con la loro madre http://baruffi.ceva.infosys.it/StoriaLoc/stormomb/STATUTI/INTRODUZIONE1.html

STATUTI E CARTE DI FRANCHIGIA DI MOMBASIGLIO Introduzione di Giovanni Olivero, " di Amedeo Michelotti e " Mombasiglio - Atlante Toponomastico del Piemonte Montano" di Fulvio Ivaldi. La pubblicazione degli statuti di Mombasiglio costituisce quindi un importante tassello nel lavoro di ricostruzione storica del nostro passato. carte di franchigia Si tratta : della vendita da parte di Ettore di Montanard, governatore di Asti, del territorio di Mombasiglio al Cardinale Giuliano della Rovere, il futuro Papa Giulio II, che lo acquista per conto del fratello Giovanni. Nel documento vengono confermati gli statuti, i patti e le antiche consuetudini del comune (8 novembre 1501); Le fonti storiche ci informano - scrive la Soprintendenza Archeologica del Piemonte - Bonifacio Giuniore, Oberto, Oddone ed Enrico dei marchesi di Ceva con la loro madre Aloisia, nel 1134, fanno donazione del castello di Mombasiglio al vescovo di Asti; solo nel 1349 Corrado di Ceva ottiene, sempre dal vescovo di Asti, l’investitura del feudo di Mombasiglio. - Sebastiano de Sauli dei marchesi di Ceva, signore di Bagnasco, Mombasiglio etc. (carta di franchigia datata Bagnasco, 24 settembre 1516);

30. Ceva's Theorem - Wikipedia, The Free Encyclopedia the theorem states that lines AD, BE and CF are concurrent if and only if.\frac{AF}{FB} \cdot \frac{BD}{. It was first proven by giovanni ceva. http://en.wikipedia.org/wiki/Ceva's_Theorem

Wikimedia needs your help in its 21-day fund drive. See our fundraising page Over US$145,000 has been donated since the drive began on 19 August. Thank you for your generosity! Ceva's theorem From Wikipedia, the free encyclopedia. (Redirected from Ceva's Theorem Ceva's theorem, case 1: the three lines are concurrent at a point O outside ABC Ceva's theorem (pronounced "Cheva") is a very popular theorem in elementary geometry . Given a triangle ABC , and points D E , and F that lie on lines BC CA , and AB respectively, the theorem states that lines AD BE and CF are concurrent if and only if It was first proven by Giovanni Ceva in his work De lineis rectis edit Proof Ceva's theorem, case 2: the three lines are concurrent at a point O inside ABC Suppose A D B E and C F intersect at a point O . Because and have the same height, we have Similarly, From this it follows that Similarly, and Multiplying these three equations gives as required. Conversely, suppose that the points D E and F satisfy the above equality. Let A D and B E intersect at O , and let C O intersect A B at F . By the direction we have just proven

31. Ceva's Theorem -- From MathWorld This theorem was first published by giovanni ceva 1678. Let PV_1, ,V_n bean arbitrary ngon, C a given point, and k a positive integer such that http://mathworld.wolfram.com/CevasTheorem.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Geometry Line Geometry Incidence Ceva's Theorem Given a triangle with polygon vertices , and and points along the sides , and , a necessary and sufficient condition for the cevians , and to be concurrent intersect in a single point) is that This theorem was first published by Giovanni Ceva 1678. Let be an arbitrary -gon, a given point, and a positive integer such that . For , let be the intersection of the lines and , then Here, and is the ratio of the lengths and Another form of the theorem is that three concurrent lines from the polygon vertices of a triangle divide the opposite sides in such fashion that the product of three nonadjacent segments equals the product of the other three (Johnson 1929, p. 147). SEE ALSO: Hoehn's Theorem Menelaus' Theorem [Pages Linking Here] REFERENCES: Beyer, W. H. (Ed.).

32. Ceva's Theorem: A Matter Of Appreciation An elegant theorem has been published by giovanni ceva in 1678. Menelaus ofAlexandria worked in the 1st century AD, giovanni ceva (16481734) was an http://www.cut-the-knot.org/Generalization/CevaPlus.shtml

Username: Password: Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents Cut The Knot! An interactive column using Java applets by Alex Bogomolny A Matter of Appreciation October 1999 I have a recollection. Years ago, a childhood friend of mine, Boris, shared with me with excitement an unusual experience he had on a visit to the Tretj'yakov Art Gallery in Moscow. He was accompanied by a professional painter, a good acquaintance of his older sister. While Boris was making a round in one of the halls, he observed that the painter remained all that time on the same spot studying a certain picture. Curious, my friend asked the painter what was it about the picture that kept him interested in it for so long. According to Boris, the painter did not reply directly, but, instead, stepped over to the picture and covered a spot on the picture with a palm of his hand. "Have a look at the picture and think of what you see," he requested. After a while, he uncovered the spot, stepped back and asked Boris to have another look. Well, almost 4 decades later, with the names of the painter and the picture long forgotten, I still vividly remember Boris' excitement when he told me of how entirely different, deeper and more beautiful, the picture appeared to him then.

33. Ceva's Theorem giovanni ceva (16481734) proved a theorem bearing his name that is seldommentioned in Elementary Geometry courses. It s a regrettable fact because not http://www.cut-the-knot.org/Generalization/ceva.shtml

Username: Password: Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents Ceva's Theorem Giovanni Ceva (1648-1734) proved a theorem bearing his name that is seldom mentioned in Elementary Geometry courses. It's a regrettable fact because not only it unifies several other more fortunate statements but its proof is actually as simple as that of the less general theorems. Additionally, the general approach affords, as is often the case, rich grounds for further meaningful explorations. Ceva's Theorem In a triangle ABC, three lines AD, BE and CF intersect at a single point K if and only if (The lines that meet at a point are said to be concurrent Proof 1 Extend the lines BE and CF beyond the triangle until they meet GH, the line through A parallel to BC. There are several pairs of similar triangles: AHF and BCF, AEG and BCE, AGK and BDK, CDK and AHK. From these and in that order we derive the following proportions: AF/FB=AH/BC (*) CE/EA=BC/AG (*) AG/BD=AK/DK AH/DC=AK/DK from the last two we conclude that AG/BD = AH/DC and, hence, BD/DC = AG/AH (*).

34. Ceva's Theorem Who is giovanni ceva? The usual proof of ceva s Theorem involves considerationof similar triangles in the augmented figure below. http://jwilson.coe.uga.edu/Texts.Folder/ratio/Ceva.html

Ceva's Theorem by Jim Wilson Given any triangle ABC with a point M in the interior. Segments through M from each vertex to the opposite sides of the triangle are Cevians and Ceva's theorem says that the product of the ratios of the pairs of segments formed on each side of the triangle by the intersection point is equal to 1, where the ratios are taken in same orientation on each side. Further, if the ratio formed by any three Cevians is equal to 1, then the three Cevians are concurrent. That is: Who is Giovanni Ceva The usual proof of Ceva's Theorem involves consideration of similar triangles in the augmented figure below. Return to the discussion

35. Ceva's Theorem giovanni ceva founded a theroem that states given a triangle ABC, the lines AE,BF, CG intersect at a single point M if and only if the following http://jwilson.coe.uga.edu/EMT669/Student.Folders/Bailey.Heather/Essay 1/Ceva's.

An Investigation into Ceva's Theorem By: Heather Bridges Giovanni Ceva founded a theroem that states given a triangle ABC, the lines AE, BF, CG intersect at a single point M if and only if the following: (AG)(BE)(CF)=(BG)(CE)(AF) (*) It is easy to see from the equation (*) above that: (AG/BG)(BE/CE)(CF/AF) = 1 It is possible to extend the lines CF and BE until they intersect the line through point A that is parallel to the line BC at H and I. Several similar triangles can now be recognized such as: AGH ~ BGC , AFI ~ CFB , AMI ~ EMB, AMH ~ EMC. We can conclude that from AGH ~ BGC that AG/GB = AH/BC and from AFI ~ CFB that AF/FC = AI/BC. Also, AMI ~ EMB shows that AI/BE = AM/ME and AMH ~ EMC shows that AH/EC = AM/ME. Therefore we know that AI/BE = AH/ EC which gives us that BE/EC = AI/AH. Then by multiplying the proportions of the sides of triangle ABC we find: (AG/GB)(BE/EC)(CF/AF) = (AH/BC)(AI/AH)(BC/AI) = 1. Therefore the statement holds that if given triangle ABC, three lines AE, BF, CG intersect at a single point M, then (AG/BG)(BE/CE)(CF/AF) = 1. It is left to prove the theorem in the other direction. Assume M is the point where AE and CG intersect, then draw a line from B through M to line AC at point F '. So we have

36. Ceva's Trisectrix giovanni ceva (16481734), an Italian mathematician and engineer, studied thecurve for b=2. This was in origin ceva s trisectrix. http://www.2dcurves.com/sextic/sextict.html

(extended) Ceva's trisectrix sextic last updated: This sextic is a botanic curve Giovanni Ceva (1648-1734), an Italian mathematician and engineer, studied the curve for b=2. This was in origin Ceva's trisectrix The curve can be written as r = sin3 f /sin f. The trisectrix can be extended to other values for b. Ceva's trisectrix can be used for the trisection of an angle, as follows (see picture to the right). Let there be a circle C with center O. Draw a line through O which cuts C in P. Construct a point Q on the x-axis so that OP = PQ. Then Ceva's trisectrix is the collection of points M for which: M lies on the line through OP MP = PQ Now the angle OQM is three times the angle QOM. For b = 1/2, the curve is called the peanut curve For b = 1, the curve is called the double egg Its equation can also be written as: r = cos f. Some examples: For large values of parameter b, the curve approximates the quadrifolium rhodonea c=2). notes 1) Cartesian equation: (x + y = ((b+1)x - (b-1)y

37. The Galileo Project ceva, giovanni. 1. Dates Born Milano, 1647 or 1648 (DBI and Loria say probablyDecember 1647) Died Gino Loria, Per la biografia de giovanni ceva , http://galileo.rice.edu/Catalog/NewFiles/ceva_gio.html

Ceva, Giovanni 1. Dates Born: Milano, 1647 or 1648 (DBI and Loria say probably December 1647) Died: Mantua, 3 or 13 May 1734 Dateinfo: Birth Uncertain Lifespan: 2. Father Occupation: Unknown I find only that Carlo Francesco Ceva was rich and famous. I accept the information: wealthy. 3. Nationality Birth: Italian Career: Italian Death: Italian 4. Education Schooling: Pisa He received his first education in a Jesuit college in Milano. He studied then in Pisa where he was a student of D. Rossetti and A. Marchetti, both students of Borelli. There is no mention of a degree. 5. Religion Affiliation: Catholic 6. Scientific Disciplines Primary: Mathematics, Hydraulics Ceva's most important mathematical work was De lineis rectis (Milan, 1678). In this work he used the properties of the center of gravity of a system of points to obtain the relations of the segments. He also published Opuscula mathematica (Milan, 1682), Geometria motus (Bologna, 1692), De re numeraria (Mantua, 1711), and other works. Much of his mathematical work had a practical bente.g., hydraulics. This became more pronounced as the years passed. His final work, and his most important one, was Opus hydrostaticum, 1728. 7. Means of Support

38. The Galileo Project Through ceva he became a correspondent and friend of giovanni ceva and Viviani.Sources. P. Fr. Gambarana, SJ, An Account of the Life of Girolamo Saccheri http://galileo.rice.edu/Catalog/NewFiles/saccheri.html

Saccheri, Giovanni Girolamo 1. Dates Born: San Remo, Genoa, 5 Sept. 1667 Died: Milano, 25 Oct. 1733 Dateinfo: Dates Certain Lifespan: 2. Father Occupation: Lawyer Giovanni Felice Saccheri was a notary. No information on financial status. 3. Nationality Birth: Italian Career: Italian Death: Italian 4. Education Schooling: Religous Order, D.D. He entered the Jesuit novitiate in Genoa in 1685. Sent to Milan in 1690, he studied philosophy and theology at the Jesuit College of Brera. Here he was influenced to study mathematics by Tommaso Ceva. As an ordained Jesuit professed of the fourth vow, he would have had a doctorate in theology. 5. Religion Affiliation: Catholic He entered the Jesuit novitiate in 1685, and was ordained a priest in 1694 at Como. 6. Scientific Disciplines Primary: Mathematics Subordinate: Mechanics His two most important books, the Logica demonstrativa (1697), an explanation of logic more geometrico, and the Euclides ab omni naevo vindicatus (1733), were virtually forgotten until they were rescued from oblivionthe Euclides by E. Beltrami in 1889 and the Logica by G. Vailati in 1903. Much of his logical and mathematical reasoning has become part of mathematical logic and non-Euclidean geometry. In 1708 he also published Neo-statica, a work in the tradition of peripatetic statics.

39. Ceva_thm The theorem is named for giovanni ceva, an Italian mathematician who lived from1648 to 1734. The lines from each vertex to the opposite side are often http://www.pballew.net/ceva_thm.html

Ceva's Theorem Ceva's Theorem states that if three lines are drawn in a triangle from each vertex to the opposite sides (AA', BB', and CC' in the figure) they intersect in a single point if, and only if, the sides are divided into parts so that :    The theorem is named for Giovanni Ceva, an Italian mathematician who lived from 1648 to 1734.  The lines from each vertex to the opposite side are often called Cevians in his honor.  You can find a biography of Ceva at the St. Andrews University web site. This theorem makes some of the geometric proofs  of concurrency almost trivial corollarys .   The medians, for example, divide each side into a 1:1 ratio, so that all three of the ratios in the formula equal 1, and therefore have a product of one.  It is almost as easy to prove the angle bisectors meet in a single point with Ceva's theorem. Here you can find a clever javascript proof of Ceva's Thm that requires nothing beyond middle school geometry formulas. There is a second simple identity that is known, but not WELL known. Let three cevians be drawn from the vertices (A, B, and C)through a common point, P, and intersecting the opposite sides (perhaps extended) at A', B', and C' as in the figure. Then for the points as described, it is true that AP/AA' + BP/BB' + CP/CC' = 2 . I was first exposed to this pretty little property in a note to the MathForum Geometry discussion list by the Greek Mathematician Antreas P. Hatzipolakis. I recently learned on one of the geometry discussion lists at the Math Forum that the Cevian is also used in 3-D for the segment from a vertex of a tetrahedron to the opposite face (possibly extended). In the same thread I had speculated that I thought the property above would extend to the tetrahedron as well with a sum of the ratios equal to three. Eisso J Atzema of The University of Maine confirmed my belief with a simple proof that extended from triangles to any N-dimensional simplex. I quote directly from his post:

40. Origins Of Some Arithmetic Terms around the end of the 19th century to honor the Italian giovanni ceva (1650? word from the surname of the Italian mathematician giovanni ceva (1647? http://www.pballew.net/arithme1.html

Origins of some Math terms Back to Math Words Alphabetical Index Abscissa is the formal term for the x-coordinate of a point on a coordinate graph. The abscissa of the point (3,5) is three. The word is a conjunction of ab (remove) + scindere (tear). Literally then, to tear or cut apart, as a line perpendicular to the x-axis would do to the coordinate plane. The main root is closely related to the Latin root from which we get the word scissors. Leibniz apparently coined the mathematical use of the term around 1692. Absolute Value The word absolute is from a variant of absolve and has a meaning related to free from restriction or condition. It seems that the mathematical phrase was first used by Karl Weierstrass in reference to complex numbers. In "The Words of Mathematics", Steven Schwartzman suggests that the use of the word for real values only became common in the middle of the 20th century. For complex numbers the absolute value is also called magnitude or length of the complex number. Complex numbers are sometimes drawn as a vector using an Argand Diagram After posting a request for information to the Historia Matematica discussion group about the use of the tilde to indicate absolute difference in England I received the following update from Herbert Prinz: Acute is from the Latin word acus for needle, with derivatives generalizing to anything pointed or sharp. The root persists in the words acid (sharp taste), acupuncture (to treat with needles) and acumen (mentally sharp). An acute angle then, is one which is sharp or pointed. In mathematics we define an acute angle as one which has a measure of less than 90

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