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139-402: The catenary curve has a U-like shape, superficially similar in appearance to a parabola , which it is not. The curve appears in the design of certain types of arches and as a cross section of the catenoid —the shape assumed by a soap film bounded by two parallel circular rings. The catenary is also called the alysoid , chainette , or, particularly in the materials sciences, an example of

278-399: A + e − x a ) , {\displaystyle y=a\cosh \left({\frac {x}{a}}\right)={\frac {a}{2}}\left(e^{\frac {x}{a}}+e^{-{\frac {x}{a}}}\right),} where cosh is the hyperbolic cosine function , and where a is the distance of the lowest point above the x axis. All catenary curves are similar to each other, since changing the parameter

417-461: A , {\displaystyle {\frac {d\varphi }{ds}}={\frac {\cos ^{2}\varphi }{a}},} and eliminating φ {\displaystyle \varphi } gives the Cesàro equation κ = a s 2 + a 2 , {\displaystyle \kappa ={\frac {a}{s^{2}+a^{2}}},} where κ {\displaystyle \kappa }

556-405: A c = 0 , {\displaystyle b^{2}-4ac=0,} or, equivalently, such that a x 2 + b x y + c y 2 {\displaystyle ax^{2}+bxy+cy^{2}} is the square of a linear polynomial . The previous section shows that any parabola with the origin as vertex and the y axis as axis of symmetry can be considered as the graph of

695-409: A cosh ⁡ x a + b {\displaystyle y=a\cosh {\frac {x}{a}}+b} revolved about the x -axis. In the mathematical model the chain (or cord, cable, rope, string, etc.) is idealized by assuming that it is so thin that it can be regarded as a curve and that it is so flexible any force of tension exerted by the chain is parallel to the chain. The analysis of

834-526: A dissertation Specimen Quaestionum Philosophicarum ex Jure collectarum ( An Essay of Collected Philosophical Problems of Right ), arguing for both a theoretical and a pedagogical relationship between philosophy and law. After one year of legal studies, he was awarded his bachelor's degree in Law on 28 September 1665. His dissertation was titled De conditionibus ( On Conditions ). In early 1666, at age 19, Leibniz wrote his first book, De Arte Combinatoria ( On

973-401: A funicular . Rope statics describes catenaries in a classic statics problem involving a hanging rope. Mathematically, the catenary curve is the graph of the hyperbolic cosine function. The surface of revolution of the catenary curve, the catenoid , is a minimal surface , specifically a minimal surface of revolution . A hanging chain will assume a shape of least potential energy which

1112-477: A is equivalent to a uniform scaling of the curve. The Whewell equation for the catenary is tan ⁡ φ = s a , {\displaystyle \tan \varphi ={\frac {s}{a}},} where φ {\displaystyle \varphi } is the tangential angle and s the arc length . Differentiating gives d φ d s = cos 2 ⁡ φ

1251-399: A connection with this curve, as Apollonius had proved. The focus–directrix property of the parabola and other conic sections was mentioned in the works of Pappus . Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity. The idea that a parabolic reflector could produce an image was already well known before the invention of

1390-700: A fair part of his assigned task: when the material Leibniz had written and collected for his history of the House of Brunswick was finally published in the 19th century, it filled three volumes. Leibniz was appointed Librarian of the Herzog August Library in Wolfenbüttel , Lower Saxony , in 1691. In 1708, John Keill , writing in the journal of the Royal Society and with Newton's presumed blessing, accused Leibniz of having plagiarised Newton's calculus. Thus began

1529-526: A flexible cable so, inverted, stand the touching pieces of an arch." In 1691, Gottfried Leibniz , Christiaan Huygens , and Johann Bernoulli derived the equation in response to a challenge by Jakob Bernoulli ; their solutions were published in the Acta Eruditorum for June 1691. David Gregory wrote a treatise on the catenary in 1697 in which he provided an incorrect derivation of the correct differential equation. Leonhard Euler proved in 1744 that

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1668-405: A function f ( x ) = a x 2  with  a ≠ 0. {\displaystyle f(x)=ax^{2}{\text{ with }}a\neq 0.} For a > 0 {\displaystyle a>0} the parabolas are opening to the top, and for a < 0 {\displaystyle a<0} are opening to the bottom (see picture). From

1807-511: A geometry book as an example to explain his reasoning. If this book was copied from an infinite chain of copies, there must be some reason for the content of the book. Leibniz concluded that there must be the " monas monadum " or God. The ontological essence of a monad is its irreducible simplicity. Unlike atoms, monads possess no material or spatial character. They also differ from atoms by their complete mutual independence, so that interactions among monads are only apparent. Instead, by virtue of

1946-441: A graph of a quadratic function. This shows that these two descriptions are equivalent. They both define curves of exactly the same shape. An alternative proof can be done using Dandelin spheres . It works without calculation and uses elementary geometric considerations only (see the derivation below). Gottfried Leibniz Gottfried Wilhelm Leibniz or Leibnitz (1 July 1646 [ O.S. 21 June] – 14 November 1716)

2085-594: A journal article titled "New System of the Nature and Communication of Substances". Between 1695 and 1705, he composed his New Essays on Human Understanding , a lengthy commentary on John Locke 's 1690 An Essay Concerning Human Understanding , but upon learning of Locke's 1704 death, lost the desire to publish it, so that the New Essays were not published until 1765. The Monadologie , composed in 1714 and published posthumously, consists of 90 aphorisms. Leibniz also wrote

2224-607: A key role in advancing his mathematical and scientific reputation, which in turn enhanced his eminence in diplomacy, history, theology, and philosophy. The Elector Ernest Augustus commissioned Leibniz to write a history of the House of Brunswick, going back to the time of Charlemagne or earlier, hoping that the resulting book would advance his dynastic ambitions. From 1687 to 1690, Leibniz traveled extensively in Germany, Austria, and Italy, seeking and finding archival materials bearing on this project. Decades went by but no history appeared;

2363-475: A letter to Thomas Paine on the construction of an arch for a bridge: I have lately received from Italy a treatise on the equilibrium of arches, by the Abbé Mascheroni. It appears to be a very scientifical work. I have not yet had time to engage in it; but I find that the conclusions of his demonstrations are, that every part of the catenary is in perfect equilibrium. It is often said that Galileo thought

2502-546: A meticulous study (informed by the 1999 additions to the critical edition) of all of Leibniz's philosophical writings up to 1688, Mercer (2001) disagreed with Couturat's reading. Leibniz met Baruch Spinoza in 1676, read some of his unpublished writings, and had since been influenced by some of Spinoza's ideas. While Leibniz befriended him and admired Spinoza's powerful intellect, he was also dismayed by Spinoza's conclusions, especially when these were inconsistent with Christian orthodoxy. Unlike Descartes and Spinoza, Leibniz had

2641-489: A parabola can then be transformed by the uniform scaling ( x , y ) → ( a x , a y ) {\displaystyle (x,y)\to (ax,ay)} into the unit parabola with equation y = x 2 {\displaystyle y=x^{2}} . Thus, any parabola can be mapped to the unit parabola by a similarity. A synthetic approach, using similar triangles, can also be used to establish this result. The general result

2780-420: A parabola is the inverse of a cardioid . Remark 2: The second polar form is a special case of a pencil of conics with focus F = ( 0 , 0 ) {\displaystyle F=(0,0)} (see picture): r = p 1 − e cos ⁡ φ {\displaystyle r={\frac {p}{1-e\cos \varphi }}} ( e {\displaystyle e}

2919-409: A philosopher, he was a leading representative of 17th-century rationalism and idealism . As a mathematician, his major achievement was the development of the main ideas of differential and integral calculus , independently of Isaac Newton 's contemporaneous developments. Mathematicians have consistently favored Leibniz's notation as the conventional and more exact expression of calculus. In

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3058-410: A post he held for the rest of his life. Leibniz served three consecutive rulers of the House of Brunswick as historian, political adviser, and most consequentially, as librarian of the ducal library. He thenceforth employed his pen on all the various political, historical, and theological matters involving the House of Brunswick; the resulting documents form a valuable part of the historical record for

3197-636: A related mission to the English government in London, early in 1673. There Leibniz came into acquaintance of Henry Oldenburg and John Collins . He met with the Royal Society where he demonstrated a calculating machine that he had designed and had been building since 1670. The machine was able to execute all four basic operations (adding, subtracting, multiplying, and dividing), and the society quickly made him an external member. The mission ended abruptly when news of

3336-658: A role in the initiatives and negotiations leading up to that Act, but not always an effective one. For example, something he published anonymously in England, thinking to promote the Brunswick cause, was formally censured by the British Parliament . The Brunswicks tolerated the enormous effort Leibniz devoted to intellectual pursuits unrelated to his duties as a courtier, pursuits such as perfecting calculus, writing about other mathematics, logic, physics, and philosophy, and keeping up

3475-493: A segment using the fact that these forces must be in balance if the chain is in static equilibrium . Let the path followed by the chain be given parametrically by r = ( x , y ) = ( x ( s ), y ( s )) where s represents arc length and r is the position vector . This is the natural parameterization and has the property that d r d s = u {\displaystyle {\frac {d\mathbf {r} }{ds}}=\mathbf {u} } where u

3614-640: A set of points ( locus of points ) in the Euclidean plane: The midpoint V {\displaystyle V} of the perpendicular from the focus F {\displaystyle F} onto the directrix l {\displaystyle l} is called the vertex , and the line F V {\displaystyle FV} is the axis of symmetry of the parabola. If one introduces Cartesian coordinates , such that F = ( 0 , f ) ,   f > 0 , {\displaystyle F=(0,f),\ f>0,} and

3753-542: A short paper, "Primae veritates" ("First Truths"), first published by Louis Couturat in 1903 (pp. 518–523) summarizing his views on metaphysics . The paper is undated; that he wrote it while in Vienna in 1689 was determined only in 1999, when the ongoing critical edition finally published Leibniz's philosophical writings for the period 1677–1690. Couturat's reading of this paper influenced much 20th-century thinking about Leibniz, especially among analytic philosophers . After

3892-485: A special case of marine vehicles moving although moored by the two catenaries each of one or more cables (wire ropes or chains) passing through the vehicle and moved along by motorized sheaves. The catenaries can be evaluated graphically. The equation of a catenary in Cartesian coordinates has the form y = a cosh ⁡ ( x a ) = a 2 ( e x

4031-401: A triangle, but the catenary must have parameters corresponding to the shape and dimensions of the wheels. Over any horizontal interval, the ratio of the area under the catenary to its length equals a , independent of the interval selected. The catenary is the only plane curve other than a horizontal line with this property. Also, the geometric centroid of the area under a stretch of catenary is

4170-448: A true catenary curve.) In optics and electromagnetics, the hyperbolic cosine and sine functions are basic solutions to Maxwell's equations. The symmetric modes consisting of two evanescent waves would form a catenary shape. The word "catenary" is derived from the Latin word catēna , which means " chain ". The English word "catenary" is usually attributed to Thomas Jefferson , who wrote in

4309-663: A two-year residence in Vienna , where he was appointed Imperial Court Councillor to the Habsburgs . On the death of Queen Anne in 1714, Elector George Louis became King George I of Great Britain , under the terms of the 1701 Act of Settlement. Even though Leibniz had done much to bring about this happy event, it was not to be his hour of glory. Despite the intercession of the Princess of Wales, Caroline of Ansbach, George I forbade Leibniz to join him in London until he completed at least one volume of

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4448-462: A university education in philosophy. He was influenced by his Leipzig professor Jakob Thomasius , who also supervised his BA thesis in philosophy. Leibniz also read Francisco Suárez , a Spanish Jesuit respected even in Lutheran universities. Leibniz was deeply interested in the new methods and conclusions of Descartes, Huygens, Newton, and Boyle , but the established philosophical ideas in which he

4587-399: A vast correspondence. He began working on calculus in 1674; the earliest evidence of its use in his surviving notebooks is 1675. By 1677 he had a coherent system in hand, but did not publish it until 1684. Leibniz's most important mathematical papers were published between 1682 and 1692, usually in a journal which he and Otto Mencke founded in 1682, the Acta Eruditorum . That journal played

4726-600: A way to solve the problem of doubling the cube using parabolas. (The solution, however, does not meet the requirements of compass-and-straightedge construction .) The area enclosed by a parabola and a line segment, the so-called "parabola segment", was computed by Archimedes by the method of exhaustion in the 3rd century BC, in his The Quadrature of the Parabola . The name "parabola" is due to Apollonius , who discovered many properties of conic sections. It means "application", referring to "application of areas" concept, that has

4865-538: A wide variety of advanced philosophical and theological works—ones that he would not have otherwise been able to read until his college years. Access to his father's library, largely written in Latin , also led to his proficiency in the Latin language, which he achieved by the age of 12. At the age of 13 he composed 300 hexameters of Latin verse in a single morning for a special event at school. In April 1661 he enrolled in his father's former university at age 14. There he

5004-681: Is V = ( 0 , 0 ) {\displaystyle V=(0,0)} , and its focus is F = ( p 2 , 0 ) {\displaystyle F=\left({\tfrac {p}{2}},0\right)} . If one shifts the origin into the focus, that is, F = ( 0 , 0 ) {\displaystyle F=(0,0)} , one obtains the equation r = p 1 − cos ⁡ φ , φ ≠ 2 π k . {\displaystyle r={\frac {p}{1-\cos \varphi }},\quad \varphi \neq 2\pi k.} Remark 1: Inverting this polar form shows that

5143-421: Is tangential to the conical surface. The graph of a quadratic function y = a x 2 + b x + c {\displaystyle y=ax^{2}+bx+c} (with a ≠ 0 {\displaystyle a\neq 0} ) is a parabola with its axis parallel to the y -axis. Conversely, every such parabola is the graph of a quadratic function. The line perpendicular to

5282-446: Is U-shaped ( opening to the top ). The horizontal chord through the focus (see picture in opening section) is called the latus rectum ; one half of it is the semi-latus rectum . The latus rectum is parallel to the directrix. The semi-latus rectum is designated by the letter p {\displaystyle p} . From the picture one obtains p = 2 f . {\displaystyle p=2f.} The latus rectum

5421-431: Is a unit tangent vector . A differential equation for the curve may be derived as follows. Let c be the lowest point on the chain, called the vertex of the catenary. The slope ⁠ dy / dx ⁠ of the curve is zero at c since it is a minimum point. Assume r is to the right of c since the other case is implied by symmetry. The forces acting on the section of the chain from c to r are

5560-412: Is a catenary. Galileo Galilei in 1638 discussed the catenary in the book Two New Sciences recognizing that it was different from a parabola . The mathematical properties of the catenary curve were studied by Robert Hooke in the 1670s, and its equation was derived by Leibniz , Huygens and Johann Bernoulli in 1691. Catenaries and related curves are used in architecture and engineering (e.g., in

5699-421: Is convenient to write a = T 0 w {\displaystyle a={\frac {T_{0}}{w}}} which is the length of chain whose weight is equal in magnitude to the tension at c . Then d y d x = s a {\displaystyle {\frac {dy}{dx}}={\frac {s}{a}}} is an equation defining the curve. The horizontal component of

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5838-420: Is defined similarly for the other two conics – the ellipse and the hyperbola. The latus rectum is the line drawn through a focus of a conic section parallel to the directrix and terminated both ways by the curve. For any case, p {\displaystyle p} is the radius of the osculating circle at the vertex. For a parabola, the semi-latus rectum, p {\displaystyle p} ,

5977-475: Is exercised within natural laws, where choices are merely contingently necessary and to be decided in the event by a "wonderful spontaneity" that provides individuals with an escape from rigorous predestination. For Leibniz, "God is an absolutely perfect being". He describes this perfection later in section VI as the simplest form of something with the most substantial outcome (VI). Along these lines, he declares that every type of perfection "pertains to him (God) in

6116-410: Is force, while space , matter , and motion are merely phenomenal. He argued, against Newton, that space , time , and motion are completely relative: "As for my own opinion, I have said more than once, that I hold space to be something merely relative, as time is, that I hold it to be an order of coexistences, as time is an order of successions." Einstein, who called himself a "Leibnizian", wrote in

6255-535: Is made of an infinite number of simple substances known as monads. Monads can also be compared to the corpuscles of the mechanical philosophy of René Descartes and others. These simple substances or monads are the "ultimate units of existence in nature". Monads have no parts but still exist by the qualities that they have. These qualities are continuously changing over time, and each monad is unique. They are also not affected by time and are subject to only creation and annihilation. Monads are centers of force ; substance

6394-1063: Is not mentioned above. It is defined and discussed below, in § Position of the focus . Let us call the length of DM and of EM x , and the length of PM   y . The lengths of BM and CM are: Using the intersecting chords theorem on the chords BC and DE , we get B M ¯ ⋅ C M ¯ = D M ¯ ⋅ E M ¯ . {\displaystyle {\overline {\mathrm {BM} }}\cdot {\overline {\mathrm {CM} }}={\overline {\mathrm {DM} }}\cdot {\overline {\mathrm {EM} }}.} Substituting: 4 r y cos ⁡ θ = x 2 . {\displaystyle 4ry\cos \theta =x^{2}.} Rearranging: y = x 2 4 r cos ⁡ θ . {\displaystyle y={\frac {x^{2}}{4r\cos \theta }}.} For any given cone and parabola, r and θ are constants, but x and y are variables that depend on

6533-436: Is that two conic sections (necessarily of the same type) are similar if and only if they have the same eccentricity. Therefore, only circles (all having eccentricity 0) share this property with parabolas (all having eccentricity 1), while general ellipses and hyperbolas do not. There are other simple affine transformations that map the parabola y = a x 2 {\displaystyle y=ax^{2}} onto

6672-435: Is the chord of the parabola that is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are geometrically similar . Parabolas have the property that, if they are made of material that reflects light , then light that travels parallel to

6811-413: Is the curvature . The radius of curvature is then ρ = a sec 2 ⁡ φ , {\displaystyle \rho =a\sec ^{2}\varphi ,} which is the length of the normal between the curve and the x -axis. When a parabola is rolled along a straight line, the roulette curve traced by its focus is a catenary. The envelope of the directrix of

6950-401: Is the basis of many practical uses of parabolas. The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors and the design of ballistic missiles . It is frequently used in physics , engineering , and many other areas. The earliest known work on conic sections was by Menaechmus in the 4th century BC. He discovered

7089-408: Is the distance of the focus from the directrix. Using the parameter p {\displaystyle p} , the equation of the parabola can be rewritten as x 2 = 2 p y . {\displaystyle x^{2}=2py.} More generally, if the vertex is V = ( v 1 , v 2 ) {\displaystyle V=(v_{1},v_{2})} ,

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7228-400: Is the eccentricity). The diagram represents a cone with its axis AV . The point A is its apex . An inclined cross-section of the cone, shown in pink, is inclined from the axis by the same angle θ , as the side of the cone. According to the definition of a parabola as a conic section, the boundary of this pink cross-section EPD is a parabola. A cross-section perpendicular to the axis of

7367-444: Is transferred to a form which is then used as a guide for the placement of bricks or other building material. The Gateway Arch in St. Louis, Missouri , United States is sometimes said to be an (inverted) catenary, but this is incorrect. It is close to a more general curve called a flattened catenary, with equation y = A  cosh( Bx ) , which is a catenary if AB = 1 . While a catenary is

7506-513: The Leibniz wheel , later used in the arithmometer , the first mass-produced mechanical calculator. In philosophy and theology , Leibniz is most noted for his optimism , i.e. his conclusion that our world is, in a qualified sense, the best possible world that God could have created , a view sometimes lampooned by other thinkers, such as Voltaire in his satirical novella Candide . Leibniz, along with René Descartes and Baruch Spinoza ,

7645-506: The Royal Society that he had solved the problem of the optimal shape of an arch, and in 1675 published an encrypted solution as a Latin anagram in an appendix to his Description of Helioscopes, where he wrote that he had found "a true mathematical and mechanical form of all manner of Arches for Building." He did not publish the solution to this anagram in his lifetime, but in 1705 his executor provided it as ut pendet continuum flexile, sic stabit contiguum rigidum inversum , meaning "As hangs

7784-574: The Thirty Years' War had left German-speaking Europe exhausted, fragmented, and economically backward. Leibniz proposed to protect German-speaking Europe by distracting Louis as follows: France would be invited to take Egypt as a stepping stone towards an eventual conquest of the Dutch East Indies . In return, France would agree to leave Germany and the Netherlands undisturbed. This plan obtained

7923-547: The Théodicée of 1710 was published in his lifetime. Leibniz dated his beginning as a philosopher to his Discourse on Metaphysics , which he composed in 1686 as a commentary on a running dispute between Nicolas Malebranche and Antoine Arnauld . This led to an extensive correspondence with Arnauld; it and the Discourse were not published until the 19th century. In 1695, Leibniz made his public entrée into European philosophy with

8062-454: The calculus priority dispute which darkened the remainder of Leibniz's life. A formal investigation by the Royal Society (in which Newton was an unacknowledged participant), undertaken in response to Leibniz's demand for a retraction, upheld Keill's charge. Historians of mathematics writing since 1900 or so have tended to acquit Leibniz, pointing to important differences between Leibniz's and Newton's versions of calculus. In 1712, Leibniz began

8201-871: The eccentricity . If p > 0 , the parabola with equation y 2 = 2 p x {\displaystyle y^{2}=2px} (opening to the right) has the polar representation r = 2 p cos ⁡ φ sin 2 ⁡ φ , φ ∈ [ − π 2 , π 2 ] ∖ { 0 } {\displaystyle r=2p{\frac {\cos \varphi }{\sin ^{2}\varphi }},\quad \varphi \in \left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]\setminus \{0\}} where r 2 = x 2 + y 2 ,   x = r cos ⁡ φ {\displaystyle r^{2}=x^{2}+y^{2},\ x=r\cos \varphi } . Its vertex

8340-410: The principle of sufficient reason . Using the principle of reasoning, Leibniz concluded that the first reason of all things is God. All that we see and experience is subject to change, and the fact that this world is contingent can be explained by the possibility of the world being arranged differently in space and time. The contingent world must have some necessary reason for its existence. Leibniz uses

8479-567: The reflecting telescope . Designs were proposed in the early to mid-17th century by many mathematicians , including René Descartes , Marin Mersenne , and James Gregory . When Isaac Newton built the first reflecting telescope in 1668, he skipped using a parabolic mirror because of the difficulty of fabrication, opting for a spherical mirror . Parabolic mirrors are used in most modern reflecting telescopes and in satellite dishes and radar receivers. A parabola can be defined geometrically as

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8618-410: The 20th century, Leibniz's notions of the law of continuity and transcendental law of homogeneity found a consistent mathematical formulation by means of non-standard analysis . He was also a pioneer in the field of mechanical calculators . While working on adding automatic multiplication and division to Pascal's calculator , he was the first to describe a pinwheel calculator in 1685 and invented

8757-521: The British. Thus Leibniz went to Paris in 1672. Soon after arriving, he met Dutch physicist and mathematician Christiaan Huygens and realised that his own knowledge of mathematics and physics was patchy. With Huygens as his mentor, he began a program of self-study that soon pushed him to making major contributions to both subjects, including discovering his version of the differential and integral calculus . He met Nicolas Malebranche and Antoine Arnauld ,

8896-596: The Combinatorial Art ), the first part of which was also his habilitation thesis in Philosophy, which he defended in March 1666. De Arte Combinatoria was inspired by Ramon Llull 's Ars Magna and contained a proof of the existence of God , cast in geometrical form, and based on the argument from motion . His next goal was to earn his license and Doctorate in Law, which normally required three years of study. In 1666,

9035-569: The Elector in the hope of obtaining employment. The stratagem worked; the Elector asked Leibniz to assist with the redrafting of the legal code for the Electorate. In 1669, Leibniz was appointed assessor in the Court of Appeal. Although von Boyneburg died late in 1672, Leibniz remained under the employment of his widow until she dismissed him in 1674. Von Boyneburg did much to promote Leibniz's reputation, and

9174-664: The Elector's cautious support. In 1672, the French government invited Leibniz to Paris for discussion, but the plan was soon overtaken by the outbreak of the Franco-Dutch War and became irrelevant. Napoleon's failed invasion of Egypt in 1798 can be seen as an unwitting, late implementation of Leibniz's plan, after the Eastern hemisphere colonial supremacy in Europe had already passed from the Dutch to

9313-437: The Elector's death (12 February 1673) reached them. Leibniz promptly returned to Paris and not, as had been planned, to Mainz. The sudden deaths of his two patrons in the same winter meant that Leibniz had to find a new basis for his career. In this regard, a 1669 invitation from Duke John Frederick of Brunswick to visit Hanover proved to have been fateful. Leibniz had declined the invitation, but had begun corresponding with

9452-665: The Russian Tsar Peter the Great stopped in Bad Pyrmont and met Leibniz, who took interest in Russian matters since 1708 and was appointed advisor in 1711. Leibniz died in Hanover in 1716. At the time, he was so out of favor that neither George I (who happened to be near Hanover at that time) nor any fellow courtier other than his personal secretary attended the funeral. Even though Leibniz

9591-827: The University of Leipzig turned down Leibniz's doctoral application and refused to grant him a Doctorate in Law, most likely due to his relative youth. Leibniz subsequently left Leipzig. Leibniz then enrolled in the University of Altdorf and quickly submitted a thesis, which he had probably been working on earlier in Leipzig. The title of his thesis was Disputatio Inauguralis de Casibus Perplexis in Jure ( Inaugural Disputation on Ambiguous Legal Cases ). Leibniz earned his license to practice law and his Doctorate in Law in November 1666. He next declined

9730-407: The apex of the cone, D and E move along the parabola, always maintaining the relationship between x and y shown in the equation. The parabolic curve is therefore the locus of points where the equation is satisfied, which makes it a Cartesian graph of the quadratic function in the equation. It is proved in a preceding section that if a parabola has its vertex at the origin, and if it opens in

9869-449: The arbitrary height at which the horizontal cross-section BECD is made. This last equation shows the relationship between these variables. They can be interpreted as Cartesian coordinates of the points D and E, in a system in the pink plane with P as its origin. Since x is squared in the equation, the fact that D and E are on opposite sides of the y axis is unimportant. If the horizontal cross-section moves up or down, toward or away from

10008-403: The axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel (" collimated ") beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other waves . This reflective property

10147-416: The best of all masters" and he will know when his good succeeds, so we, therefore, must act in conformity to his good will—or as much of it as we understand (IV). In our view of God, Leibniz declares that we cannot admire the work solely because of the maker, lest we mar the glory and love God in doing so. Instead, we must admire the maker for the work he has done (II). Effectively, Leibniz states that if we say

10286-404: The catenary is the curve which, when rotated about the x -axis, gives the surface of minimum surface area (the catenoid ) for the given bounding circles. Nicolas Fuss gave equations describing the equilibrium of a chain under any force in 1796. Catenary arches are often used in the construction of kilns . To create the desired curve, the shape of a hanging chain of the desired dimensions

10425-414: The chain, and so the chain follows the catenary curve. The same is true of a simple suspension bridge or "catenary bridge," where the roadway follows the cable. A stressed ribbon bridge is a more sophisticated structure with the same catenary shape. However, in a suspension bridge with a suspended roadway, the chains or cables support the weight of the bridge, and so do not hang freely. In most cases

10564-625: The coming of the Industrial Revolution and the spread of specialized labor. He is a prominent figure in both the history of philosophy and the history of mathematics . He wrote works on philosophy , theology , ethics , politics , law , history , philology , games , music , and other studies. Leibniz also made major contributions to physics and technology , and anticipated notions that surfaced much later in probability theory , biology , medicine , geology , psychology , linguistics and computer science . Leibniz contributed to

10703-399: The cone passes through the vertex P of the parabola. This cross-section is circular, but appears elliptical when viewed obliquely, as is shown in the diagram. Its centre is V, and PK is a diameter. We will call its radius  r . Another perpendicular to the axis, circular cross-section of the cone is farther from the apex A than the one just described. It has a chord DE , which joins

10842-401: The consort of her grandson, the future George II . To each of these women he was correspondent, adviser, and friend. In turn, they all approved of Leibniz more than did their spouses and the future king George I of Great Britain . The population of Hanover was only about 10,000, and its provinciality eventually grated on Leibniz. Nevertheless, to be a major courtier to the House of Brunswick

10981-400: The curve for an optimal arch is similar except that the forces of tension become forces of compression and everything is inverted. An underlying principle is that the chain may be considered a rigid body once it has attained equilibrium. Equations which define the shape of the curve and the tension of the chain at each point may be derived by a careful inspection of the various forces acting on

11120-426: The curve of a hanging chain was parabolic. However, in his Two New Sciences (1638), Galileo wrote that a hanging cord is only an approximate parabola, correctly observing that this approximation improves in accuracy as the curvature gets smaller and is almost exact when the elevation is less than 45°. The fact that the curve followed by a chain is not a parabola was proven by Joachim Jungius (1587–1657); this result

11259-436: The design of bridges and arches so that forces do not result in bending moments). In the offshore oil and gas industry, "catenary" refers to a steel catenary riser , a pipeline suspended between a production platform and the seabed that adopts an approximate catenary shape. In the rail industry it refers to the overhead wiring that transfers power to trains. (This often supports a contact wire, in which case it does not follow

11398-415: The directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the " vertex " and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The " latus rectum "

11537-513: The directrix has the equation y = − 1 4 {\displaystyle y=-{\tfrac {1}{4}}} . The general function of degree 2 is f ( x ) = a x 2 + b x + c      with      a , b , c ∈ R ,   a ≠ 0. {\displaystyle f(x)=ax^{2}+bx+c~~{\text{ with }}~~a,b,c\in \mathbb {R} ,\ a\neq 0.} Completing

11676-752: The directrix has the equation y = − f {\displaystyle y=-f} , one obtains for a point P = ( x , y ) {\displaystyle P=(x,y)} from | P F | 2 = | P l | 2 {\displaystyle |PF|^{2}=|Pl|^{2}} the equation x 2 + ( y − f ) 2 = ( y + f ) 2 {\displaystyle x^{2}+(y-f)^{2}=(y+f)^{2}} . Solving for y {\displaystyle y} yields y = 1 4 f x 2 . {\displaystyle y={\frac {1}{4f}}x^{2}.} This parabola

11815-557: The duke in 1671. In 1673, the duke offered Leibniz the post of counsellor. Leibniz very reluctantly accepted the position two years later, only after it became clear that no employment was forthcoming in Paris, whose intellectual stimulation he relished, or with the Habsburg imperial court. In 1675 he tried to get admitted to the French Academy of Sciences as a foreign honorary member, but it

11954-661: The equation uses the Hesse normal form of a line to calculate the distance | P l | {\displaystyle |Pl|} ). For a parametric equation of a parabola in general position see § As the affine image of the unit parabola . The implicit equation of a parabola is defined by an irreducible polynomial of degree two: a x 2 + b x y + c y 2 + d x + e y + f = 0 , {\displaystyle ax^{2}+bxy+cy^{2}+dx+ey+f=0,} such that b 2 − 4

12093-579: The exercise of their free will . God does not arbitrarily inflict pain and suffering on humans; rather he permits both moral evil (sin) and physical evil (pain and suffering) as the necessary consequences of metaphysical evil (imperfection), as a means by which humans can identify and correct their erroneous decisions, and as a contrast to true good. Further, although human actions flow from prior causes that ultimately arise in God and therefore are known to God as metaphysical certainties, an individual's free will

12232-571: The fair sum he left to his sole heir, his sister's stepson, proved that the Brunswicks had paid him fairly well. In his diplomatic endeavors, he at times verged on the unscrupulous, as was often the case with professional diplomats of his day. On several occasions, Leibniz backdated and altered personal manuscripts, actions which put him in a bad light during the calculus controversy . He was charming, well-mannered, and not without humor and imagination. He had many friends and admirers all over Europe. He

12371-436: The field of library science by developing a cataloguing system while working at the Herzog August Library in Wolfenbüttel , Germany, that served as a model for many of Europe's largest libraries. His contributions to a wide range of subjects were scattered in various learned journals , in tens of thousands of letters and in unpublished manuscripts. He wrote in several languages, primarily in Latin, French and German. As

12510-752: The focus F = ( v 1 , v 2 + f ) {\displaystyle F=(v_{1},v_{2}+f)} , and the directrix y = v 2 − f {\displaystyle y=v_{2}-f} , one obtains the equation y = 1 4 f ( x − v 1 ) 2 + v 2 = 1 4 f x 2 − v 1 2 f x + v 1 2 4 f + v 2 . {\displaystyle y={\frac {1}{4f}}(x-v_{1})^{2}+v_{2}={\frac {1}{4f}}x^{2}-{\frac {v_{1}}{2f}}x+{\frac {v_{1}^{2}}{4f}}+v_{2}.} Remarks : If

12649-638: The focus is F = ( f 1 , f 2 ) {\displaystyle F=(f_{1},f_{2})} , and the directrix a x + b y + c = 0 {\displaystyle ax+by+c=0} , then one obtains the equation ( a x + b y + c ) 2 a 2 + b 2 = ( x − f 1 ) 2 + ( y − f 2 ) 2 {\displaystyle {\frac {(ax+by+c)^{2}}{a^{2}+b^{2}}}=(x-f_{1})^{2}+(y-f_{2})^{2}} (the left side of

12788-422: The highest degree" (I). Even though his types of perfections are not specifically drawn out, Leibniz highlights the one thing that, to him, does certify imperfections and proves that God is perfect: "that one acts imperfectly if he acts with less perfection than he is capable of", and since God is a perfect being, he cannot act imperfectly (III). Because God cannot act imperfectly, the decisions he makes pertaining to

12927-521: The history of the Brunswick family his father had commissioned nearly 30 years earlier. Moreover, for George I to include Leibniz in his London court would have been deemed insulting to Newton, who was seen as having won the calculus priority dispute and whose standing in British official circles could not have been higher. Finally, his dear friend and defender, the Dowager Electress Sophia, died in 1714. In 1716, while traveling in northern Europe,

13066-468: The ideal shape for a freestanding arch of constant thickness, the Gateway Arch is narrower near the top. According to the U.S. National Historic Landmark nomination for the arch, it is a " weighted catenary " instead. Its shape corresponds to the shape that a weighted chain, having lighter links in the middle, would form. In free-hanging chains, the force exerted is uniform with respect to length of

13205-482: The introduction to Max Jammer 's book Concepts of Space that Leibnizianism was superior to Newtonianism, and his ideas would have dominated over Newton's had it not been for the poor technological tools of the time; Joseph Agassi argues that Leibniz paved the way for Einstein's theory of relativity . Leibniz's proof of God can be summarized in the Théodicée . Reason is governed by the principle of contradiction and

13344-499: The journey from London to Hanover, Leibniz stopped in The Hague where he met van Leeuwenhoek , the discoverer of microorganisms. He also spent several days in intense discussion with Spinoza , who had just completed, but had not published, his masterwork, the Ethics . Spinoza died very shortly after Leibniz's visit. In 1677, he was promoted, at his request, to Privy Counselor of Justice,

13483-611: The latter's memoranda and letters began to attract favorable notice. After Leibniz's service to the Elector there soon followed a diplomatic role. He published an essay, under the pseudonym of a fictitious Polish nobleman, arguing (unsuccessfully) for the German candidate for the Polish crown. The main force in European geopolitics during Leibniz's adult life was the ambition of Louis XIV of France , backed by French military and economic might. Meanwhile,

13622-400: The leading French philosophers of the day, and studied the writings of Descartes and Pascal , unpublished as well as published. He befriended a German mathematician, Ehrenfried Walther von Tschirnhaus ; they corresponded for the rest of their lives. When it became clear that France would not implement its part of Leibniz's Egyptian plan, the Elector sent his nephew, escorted by Leibniz, on

13761-400: The midpoint of the perpendicular segment connecting the centroid of the curve itself and the x -axis. A moving charge in a uniform electric field travels along a catenary (which tends to a parabola if the charge velocity is much less than the speed of light c ). The surface of revolution with fixed radii at either end that has minimum surface area is a catenary y =

13900-496: The next Elector became quite annoyed at Leibniz's apparent dilatoriness. Leibniz never finished the project, in part because of his huge output on many other fronts, but also because he insisted on writing a meticulously researched and erudite book based on archival sources, when his patrons would have been quite happy with a short popular book, one perhaps little more than a genealogy with commentary, to be completed in three years or less. They never knew that he had in fact carried out

14039-483: The offer of an academic appointment at Altdorf, saying that "my thoughts were turned in an entirely different direction". As an adult, Leibniz often introduced himself as "Gottfried von Leibniz". Many posthumously published editions of his writings presented his name on the title page as " Freiherr G. W. von Leibniz." However, no document has ever been found from any contemporary government that stated his appointment to any form of nobility . Leibniz's first position

14178-510: The only way we can truly love God is by being content "with all that comes to us according to his will" (IV). Because God is "an absolutely perfect being" (I), Leibniz argues that God would be acting imperfectly if he acted with any less perfection than what he is able of (III). His syllogism then ends with the statement that God has made the world perfectly in all ways. This also affects how we should view God and his will. Leibniz states that, in lieu of God's will, we have to understand that God "is

14317-398: The origin (0, 0) and the same semi-latus rectum p {\displaystyle p} can be represented by the equation y 2 = 2 p x + ( e 2 − 1 ) x 2 , e ≥ 0 , {\displaystyle y^{2}=2px+(e^{2}-1)x^{2},\quad e\geq 0,} with e {\displaystyle e}

14456-430: The origin as vertex. A suitable rotation around the origin can then transform the parabola to one that has the y axis as axis of symmetry. Hence the parabola P {\displaystyle {\mathcal {P}}} can be transformed by a rigid motion to a parabola with an equation y = a x 2 ,   a ≠ 0 {\displaystyle y=ax^{2},\ a\neq 0} . Such

14595-587: The other by a similarity , that is, an arbitrary composition of rigid motions ( translations and rotations ) and uniform scalings . A parabola P {\displaystyle {\mathcal {P}}} with vertex V = ( v 1 , v 2 ) {\displaystyle V=(v_{1},v_{2})} can be transformed by the translation ( x , y ) → ( x − v 1 , y − v 2 ) {\displaystyle (x,y)\to (x-v_{1},y-v_{2})} to one with

14734-448: The parabola is also a catenary. The involute from the vertex, that is the roulette traced by a point starting at the vertex when a line is rolled on a catenary, is the tractrix . Another roulette, formed by rolling a line on a catenary, is another line. This implies that square wheels can roll perfectly smoothly on a road made of a series of bumps in the shape of an inverted catenary curve. The wheels can be any regular polygon except

14873-403: The parabola, which is the distance from the vertex to the focus. The focus and the point F are therefore equally distant from the vertex, along the same line, which implies that they are the same point. Therefore, the point F, defined above, is the focus of the parabola . This discussion started from the definition of a parabola as a conic section, but it has now led to a description as

15012-625: The period. Leibniz began promoting a project to use windmills to improve the mining operations in the Harz Mountains. This project did little to improve mining operations and was shut down by Duke Ernst August in 1685. Among the few people in north Germany to accept Leibniz were the Electress Sophia of Hanover (1630–1714), her daughter Sophia Charlotte of Hanover (1668–1705), the Queen of Prussia and his avowed disciple, and Caroline of Ansbach ,

15151-405: The perpendicular from the point V to the plane of the parabola. By symmetry, F is on the axis of symmetry of the parabola. Angle VPF is complementary to θ , and angle PVF is complementary to angle VPF, therefore angle PVF is θ . Since the length of PV is r , the distance of F from the vertex of the parabola is r sin θ . It is shown above that this distance equals the focal length of

15290-399: The points where the parabola intersects the circle. Another chord BC is the perpendicular bisector of DE and is consequently a diameter of the circle. These two chords and the parabola's axis of symmetry PM all intersect at the point M. All the labelled points, except D and E, are coplanar . They are in the plane of symmetry of the whole figure. This includes the point F, which

15429-407: The positive y direction, then its equation is y = ⁠ x / 4 f ⁠ , where f is its focal length. Comparing this with the last equation above shows that the focal length of the parabola in the cone is r sin θ . In the diagram above, the point V is the foot of the perpendicular from the vertex of the parabola to the axis of the cone. The point F is the foot of

15568-426: The principle of pre-established harmony , each monad follows a pre-programmed set of "instructions" peculiar to itself, so that a monad "knows" what to do at each moment. By virtue of these intrinsic instructions, each monad is like a little mirror of the universe. Monads need not be "small"; e.g., each human being constitutes a monad, in which case free will is problematic. Monads are purported to have gotten rid of

15707-531: The problematic: The Theodicy tries to justify the apparent imperfections of the world by claiming that it is optimal among all possible worlds . It must be the best possible and most balanced world, because it was created by an all powerful and all knowing God, who would not choose to create an imperfect world if a better world could be known to him or possible to exist. In effect, apparent flaws that can be identified in this world must exist in every possible world, because otherwise God would have chosen to create

15846-670: The roadway is flat, so when the weight of the cable is negligible compared with the weight being supported, the force exerted is uniform with respect to horizontal distance, and the result is a parabola , as discussed below (although the term "catenary" is often still used, in an informal sense). If the cable is heavy then the resulting curve is between a catenary and a parabola. The catenary produced by gravity provides an advantage to heavy anchor rodes . An anchor rode (or anchor line) usually consists of chain or cable or both. Anchor rodes are used by ships, oil rigs, docks, floating wind turbines , and other marine equipment which must be anchored to

15985-437: The same curves. One description of a parabola involves a point (the focus ) and a line (the directrix ). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from the directrix and the focus. Another description of a parabola is as a conic section , created from the intersection of a right circular conical surface and a plane parallel to another plane that

16124-541: The seabed. When the rope is slack, the catenary curve presents a lower angle of pull on the anchor or mooring device than would be the case if it were nearly straight. This enhances the performance of the anchor and raises the level of force it will resist before dragging. To maintain the catenary shape in the presence of wind, a heavy chain is needed, so that only larger ships in deeper water can rely on this effect. Smaller boats also rely on catenary to maintain maximum holding power. Cable ferries and chain boats present

16263-470: The section above one obtains: For a = 1 {\displaystyle a=1} the parabola is the unit parabola with equation y = x 2 {\displaystyle y=x^{2}} . Its focus is ( 0 , 1 4 ) {\displaystyle \left(0,{\tfrac {1}{4}}\right)} , the semi-latus rectum p = 1 2 {\displaystyle p={\tfrac {1}{2}}} , and

16402-399: The section to the right. The tension at r can be split into two components so it may be written T u = ( T cos φ , T sin φ ) , where T is the magnitude of the force and φ is the angle between the curve at r and the x -axis (see tangential angle ). Finally, the weight of the chain is represented by (0, − ws ) where w is the weight per unit length and s is the length of

16541-621: The segment of chain between c and r . The chain is in equilibrium so the sum of three forces is 0 , therefore T cos ⁡ φ = T 0 {\displaystyle T\cos \varphi =T_{0}} and T sin ⁡ φ = w s , {\displaystyle T\sin \varphi =ws\,,} and dividing these gives d y d x = tan ⁡ φ = w s T 0 . {\displaystyle {\frac {dy}{dx}}=\tan \varphi ={\frac {ws}{T_{0}}}\,.} It

16680-425: The square yields f ( x ) = a ( x + b 2 a ) 2 + 4 a c − b 2 4 a , {\displaystyle f(x)=a\left(x+{\frac {b}{2a}}\right)^{2}+{\frac {4ac-b^{2}}{4a}},} which is the equation of a parabola with Two objects in the Euclidean plane are similar if one can be transformed to

16819-399: The tension of the chain at c , the tension of the chain at r , and the weight of the chain. The tension at c is tangent to the curve at c and is therefore horizontal without any vertical component and it pulls the section to the left so it may be written (− T 0 , 0) where T 0 is the magnitude of the force. The tension at r is parallel to the curve at r and pulls

16958-425: The tension, T cos φ = T 0 is constant and the vertical component of the tension, T sin φ = ws is proportional to the length of chain between r and the vertex. Parabola In mathematics , a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly

17097-518: The term " possible world " to define modal notions. Gottfried Leibniz was born on July 1 [ OS : June 21], 1646, in Leipzig , Saxony, to Friedrich Leibniz (1597–1652) and Catharina Schmuck (1621–1664). He was baptized two days later at St. Nicholas Church, Leipzig ; his godfather was the Lutheran theologian Martin Geier  [ de ] . His father died when he was six years old, and Leibniz

17236-435: The unit parabola, such as ( x , y ) → ( x , y a ) {\displaystyle (x,y)\to \left(x,{\tfrac {y}{a}}\right)} . But this mapping is not a similarity, and only shows that all parabolas are affinely equivalent (see § As the affine image of the unit parabola ). The pencil of conic sections with the x axis as axis of symmetry, one vertex at

17375-602: The view of Leibniz, because reason and faith must be entirely reconciled, any tenet of faith which could not be defended by reason must be rejected. Leibniz then approached one of the central criticisms of Christian theism: if God is all good , all wise , and all powerful , then how did evil come into the world ? The answer (according to Leibniz) is that, while God is indeed unlimited in wisdom and power, his human creations, as creations, are limited both in their wisdom and in their will (power to act). This predisposes humans to false beliefs, wrong decisions, and ineffective actions in

17514-423: The world must be perfect. Leibniz also comforts readers, stating that because he has done everything to the most perfect degree; those who love him cannot be injured. However, to love God is a subject of difficulty as Leibniz believes that we are "not disposed to wish for that which God desires" because we have the ability to alter our disposition (IV). In accordance with this, many act as rebels, but Leibniz says that

17653-406: The world that excluded those flaws. Leibniz asserted that the truths of theology (religion) and philosophy cannot contradict each other, since reason and faith are both "gifts of God" so that their conflict would imply God contending against himself. The Theodicy is Leibniz's attempt to reconcile his personal philosophical system with his interpretation of the tenets of Christianity. This project

17792-450: Was a German polymath active as a mathematician , philosopher , scientist and diplomat who is credited, alongside Sir Isaac Newton , with the invention of calculus in addition to many other branches of mathematics , such as binary arithmetic, and statistics . Leibniz has been called the "last universal genius" due to his knowledge and skills in different fields and because such people became much less common after his lifetime with

17931-750: Was a life member of the Royal Society and the Berlin Academy of Sciences , neither organization saw fit to honor his death. His grave went unmarked for more than 50 years. He was, however, eulogized by Fontenelle , before the French Academy of Sciences in Paris, which had admitted him as a foreign member in 1700. The eulogy was composed at the behest of the Duchess of Orleans , a niece of the Electress Sophia. Leibniz never married. He proposed to an unknown woman at age 50, but changed his mind when she took too long to decide. He complained on occasion about money, but

18070-583: Was as a salaried secretary to an alchemical society in Nuremberg . He knew fairly little about the subject at that time but presented himself as deeply learned. He soon met Johann Christian von Boyneburg (1622–1672), the dismissed chief minister of the Elector of Mainz , Johann Philipp von Schönborn . Von Boyneburg hired Leibniz as an assistant, and shortly thereafter reconciled with the Elector and introduced Leibniz to him. Leibniz then dedicated an essay on law to

18209-548: Was considered that there were already enough foreigners there and so no invitation came. He left Paris in October 1676. Leibniz managed to delay his arrival in Hanover until the end of 1676 after making one more short journey to London, where Newton accused him of having seen his unpublished work on calculus in advance. This was alleged to be evidence supporting the accusation, made decades later, that he had stolen calculus from Newton. On

18348-460: Was educated influenced his view of their work. Leibniz variously invoked one or another of seven fundamental philosophical Principles: Leibniz would on occasion give a rational defense of a specific principle, but more often took them for granted. Leibniz's best known contribution to metaphysics is his theory of monads , as exposited in Monadologie . He proposes his theory that the universe

18487-651: Was guided, among others, by Jakob Thomasius , previously a student of Friedrich. Leibniz completed his bachelor's degree in Philosophy in December 1662. He defended his Disputatio Metaphysica de Principio Individui ( Metaphysical Disputation on the Principle of Individuation ), which addressed the principle of individuation , on 9 June 1663 [ O.S. 30 May], presenting an early version of monadic substance theory. Leibniz earned his master's degree in Philosophy on 7 February 1664. In December 1664 he published and defended

18626-438: Was identified as a Protestant and a philosophical theist . Leibniz remained committed to Trinitarian Christianity throughout his life. Leibniz's philosophical thinking appears fragmented because his philosophical writings consist mainly of a multitude of short pieces: journal articles, manuscripts published long after his death, and letters to correspondents. He wrote two book-length philosophical treatises, of which only

18765-656: Was motivated in part by Leibniz's belief, shared by many philosophers and theologians during the Enlightenment , in the rational and enlightened nature of the Christian religion. It was also shaped by Leibniz's belief in the perfectibility of human nature (if humanity relied on correct philosophy and religion as a guide), and by his belief that metaphysical necessity must have a rational or logical foundation, even if this metaphysical causality seemed inexplicable in terms of physical necessity (the natural laws identified by science). In

18904-403: Was one of the three influential early modern rationalists . His philosophy also assimilates elements of the scholastic tradition, notably the assumption that some substantive knowledge of reality can be achieved by reasoning from first principles or prior definitions. The work of Leibniz anticipated modern logic and still influences contemporary analytic philosophy , such as its adopted use of

19043-500: Was published posthumously in 1669. The application of the catenary to the construction of arches is attributed to Robert Hooke , whose "true mathematical and mechanical form" in the context of the rebuilding of St Paul's Cathedral alluded to a catenary. Some much older arches approximate catenaries, an example of which is the Arch of Taq-i Kisra in Ctesiphon . In 1671, Hooke announced to

19182-544: Was quite an honor, especially in light of the meteoric rise in the prestige of that House during Leibniz's association with it. In 1692, the Duke of Brunswick became a hereditary Elector of the Holy Roman Empire . The British Act of Settlement 1701 designated the Electress Sophia and her descent as the royal family of England, once both King William III and his sister-in-law and successor, Queen Anne , were dead. Leibniz played

19321-437: Was raised by his mother. Leibniz's father had been a Professor of Moral Philosophy at the University of Leipzig , where he also served as dean of philosophy. The boy inherited his father's personal library. He was given free access to it from the age of seven, shortly after his father's death. While Leibniz's schoolwork was largely confined to the study of a small canon of authorities, his father's library enabled him to study

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