More than thirty eminent scholars from nine different countries have contributed to The Cambridge History of Eighteenth-Century Philosophy – the most comprehensive and up-to-date history of the subject available in English. In contrast with most histories of philosophy and in keeping with preceding Cambridge volumes in the series, the subject is treated systematically by topic, not by individual thinker, school, or movement, thus enabling a much more historically nuanced picture of the period to be painted. As in previous titles in the series, the volume has extensive biographical and bibliographical research materials. During the eighteenth century, the dominant concept in philosophy was human nature, and so it is around this concept that the present work is centered. This allows the contributors to offer both detailed explorations of the epistemological, metaphysical, and ethical themes that continue to stand at the forefront of philosophy and to voice a critical attitude toward the historiography behind this emphasis in philosophical thought. At the same time, due attention is paid to historical context, with particular emphasis on the connections among philosophy, science, and theology. This judiciously balanced, systematic, and comprehensive account of the whole of Western philosophy during the period will be an invaluable resource for philosophers, intellectual historians, theologians, political theorists, historians of science, and literary scholars.

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The Cambridge history of eighteenth-century philosophy

ion function: (37) α : P(Γ) → (Lab[C] → Ass) (.1) α(i)( L ) = {s : <s, L> ∈ i} if L ∈ At[C] ∪ In[C] (.2) α(i)( L ) = i ∩ S if L ∈ After[C] (.3) Since α is a bijection, the inverse of which is γ, the discovery of a global invariant i ∈ P(Γ) satisfying verification condition gvc[C][p, q](i) is equivalent to the discovery of local invariants inv(L), L ∈ L a b [ C ] satisfying verification condition gvc[C][p, q]( γ(inv) ). This leads to the construction of the local verification conditions by calculus (COUSOT & COUSOT [1982]). This equivalence is of theoretical interest only since, from a practical point of view, each local invariant is simpler than the global one and the task of checking gvc[C][p, q](γ(inv)) can be decomposed into the verification of more numerous but simpler conditions, one for each local invariant. 5.2 .3 Construction of the verification conditions for local invariants To formally derive the local verification conditions from induction principle (29), we first express the operational semantics (13) in an equivalent form using program steps. From a syntactic point of view, the next elementary step Step[C][L] which will be executed when control is at point L ∈ At[C] ∪ In[C] of command C ∈ Comp is an atomic command or a test defined by cases as follows (where n ≥ 0): DEFINITION Elementary steps within a command (38) Step[C][(...((C'; C1); C2) ...; Cn)] = C' if C' is skip, X := E or X := ? (.1) Step[C][(...(((B → C' ◊ C"); C1); C2)...; Cn)] = B (.2) Step[C][(...(((B * C'); C1); C2)...; Cn)] = B (.3) Example Elementary steps of program (4) (39) For program C defined by (4) with labels (15), we have Step[C ] = [L1 ← Z := 1, L2 ← Y <> 0, L3 ← odd(Y), L4 ← Y := Y 1, L5 ← Z := Z * X, L6 ← Y := T div 2, L7 ← X := X * X]. Again from a syntactic point of view, the next label Succ[C][L] which will be reached after execution of an elementary step when control is at point L ∈ At[C] ∪ In[C] of command C ∈ Comp can be defined by cases as follows (where n ≥ 0 and (...(C1; C2)...; Cn) is the final label √ for n = 0): DEFINITION Successors of a program control point (40) Succ[C][(...((C'; C1); C2)...; Cn)] = (.1) (...(C1; C2)...; Cn) if C' is skip, X := E or X := ? Succ[C][(...(((B → C' ◊ C"); C1); C2)...; Cn) ] = (.2) [tt ← (...((C'; C1); C2)...; Cn), ff ← (...((C"; C1); C2)...; Cn)] Succ[C][(...(((B * C); C1); C2)...; Cn)] = (.3) [tt ← (...(((C; (B * C)); C1); C2)...; Cn), ff ← (...(C1; C2)...; Cn)] Example Successors of control points of program (4) (41) For program C defined by (4) with labels (15), we have Succ[C] = [L1 ← L2, L 2 ← [ t t ← L 3 , ff ← L 8 ], L3 ← [ t t ← L 4 , ff ← L 6 ], L4 ← L 5 , L5 ← L 2 , L 6 ← L7, L7 ← L2]. Now from a semantical point of view, execution of an elementary step Step[C][L] in memory state s can lead to any successor state s' ∈ NextS[C]<s, L> as follows: DEFINITION Successor states (42) NextS[C]<s, L> = {s} if Step[C][L] is skip (.1) NextS[C]<s, L> = {s[X ← E(s)]} if Step[C][L] is X := E (.2) NextS[C]<s, L> = {s[X ← d] : d ∈ D} if Step[C][L] is X := ? (.3) NextS[C]<s, L> = {s} if Step[C][L] is B (.4) Again from a semantical point of view, the next label NextL[C]<s, L> which can be reached after execution of an elementary step in configuration <s, L> of command C ∈ Comp can be defined by cases as follows: DEFINITION Successor control point (43) NextL[C]<s, L> = {Succ[C][L]} if Step[C][L] is skip, X := E or X := ? (.1) NextL[C]<s, L> = {Succ[C][L](s ∈ B)} if Step[C][L] is B (.2) The operational semantics (13) can now be given an equivalent stepwise presentation: LEMMA Stepwise presentation of the operational semantics (44) op[C] = {<<s, L>, final <s', L'>> : s ∈ S ∧ L ∈ At[C] ∪ In[C] ∧ s' ∈ NextS[C]<s, L> ∧ L' ∈ NextL[C]<s, L> } where final <s', √> = s' and otherwise final γ = γ. We have seen that a partial correctness proof of { p }C{ q } by Floyd-Naur's method consists in discovering local invariants inv ∈ Lab[C] → Ass satisfying gvc[C][p, q]( γ(inv) ). This global verification condition is equivalent to a conjunction of simpler local verification conditions as follows: THEOREM NAUR [1966], FLOYD [1967], MANNA [1969] [1971] Floyd-Naur partial (45) correctness proof method with stepwise verification conditions A partial correctness proof of { p }C{ q } by Floyd-Naur's method consists in discovering local invariants inv ∈ Lab[C] → Ass, which must be proved to satisfy the following local verification conditions : . p ⊆ inv(L) if L ∈ At[C] (.1) . inv(L) ⊆ inv(Succ[C][L]) if L ∈ At[C] ∪ In[C] ∧ Step[C][L] is skip (.2) . inv(L) ⊆ {s ∈ S : s[X ← E(s)] ∈ inv(Succ[C][L])} (.3) if L ∈ At[C] ∪ In[C] ∧ Step[C][L] is X := E . {s[X ← d] : s ∈ inv(L) ∧ d ∈ D} ⊆ inv(Succ[C][L]) (.4) if L ∈ At[C] ∪ In[C] ∧ Step[C][L] is X := ? . (inv(L) ∩ B) ⊆ inv(Succ[C][L](tt)) if L ∈ At[C] ∪ In[C] ∧ Step[C][L] is B (.5) . (inv(L) ∩ ¬B) ⊆ inv(Succ[C][L](ff)) if L ∈ At[C] ∪ In[C] ∧ Step[C][L] is B (.6) . inv(L) ⊆ q if L ∈ After[C] (.7) The verification condition (45.3) for assignment is backward. This name arises out of the fact that the postcondition inv(Succ[C][L] is back-transformed into the assertion {s ∈ S : s[X ← E(s)] ∈ inv(Succ[C][L])} written in terms of the states before assignment. Verification condition (45.4) for random assignment is forward. Verification condition (45.3) can also be given an equivalent forward form (KING [1969]) : . {s[X ← E(s)] : s ∈ inv(L)} ⊆ inv(Succ[C][L]) if L ∈ At[C]∪In[C] ∧ Step[C][L] is X:=E (.8) Proof By (29), we have to show that γ(inv(L)) satisfies gvc[C][p, q]( γ(inv(L)) ). We proceed by simplification of gvc[C][p, q]( γ(inv(L)) ) which constructively leads to local verification conditions (45): • First, (∀ s ∈ p. <s, C> ∈ γ(inv(L))) ⇔ (∀ s ∈ p. C ∈ At[C] ∪ In[C] ∧ s ∈ inv(C)) [by (36.2)] ⇔ (p ⊆ inv(C)) [by (32.2)] ⇔ (∀ L ∈ At[C]. p ⊆ inv(L)) [by (32.2)]. • Then, according to (36) and (44), the condition γ(inv)  op[C] ⊆ Γ x γ(inv) can be decomposed into a conjunction of simpler verification conditions, one for each program step: γ(inv)  op[C] ⊆ Γ x γ(inv) ⇔ {<s, L> : s ∈ inv(L) ∧ L ∈ Lab[C] {√}}  op[C] ⊆ Γ x [{<s, L> : s ∈ inv(L) ∧ L ∈ Lab[C] {√}} ∪ inv(√)] ⇔ { < < s, L> , f i n a l < s', L'> > : s ∈ i n v ( L ) ∧ L ∈ A t [ C ] ∪ I n [ C ] ∧ s' ∈ NextS[C ]<s,L> ∧ L' ∈ NextL[C ]<s,L>} ⊆ Γ x [{<s, L> : s ∈ inv(L) ∧ L ∈ Lab[C] {√}} ∪ inv(√)] ⇔ ∀ L ∈ At[C ] ∪ In[C ]. ∀ s ∈ inv(L). {final <s', L'> : s' ∈ NextS[C ]<s, L> ∧ L' ∈ NextL[C]<s, L>} ⊆ [{<s", L"> : s" ∈ inv(L") ∧ L" ∈ Lab[C] {√}} ∪ inv(√)] We go on by cases, according to (38), (40) and (42): If Step[C][L] is X := E (skip and X: = ? are handled the same way), then we have to check that : {final <s', L'> : s' ∈ {s[X ← E(s)]} ∧ L' ∈ {Succ[C][L]} } ⊆ [{<s", L"> : s" ∈ inv(L") ∧ L" ∈ Lab[C] {√}} ∪ inv(√)] ⇔ final <s[X ← E(s)], Succ[C][L]> ∈ [{<s", L"> : s" ∈ inv(L") ∧ L" ∈ Lab[C] {√}} ∪ inv(√)] ⇔ s[X ← E(s)] ∈ inv(Succ[C][L]) (and ∀ s ∈ inv(L). s[X ← E(s)] ∈ inv(Succ[C][L]) is obviously equivalent to inv(L) ⊆ {s ∈ S : s[X ← E(s)] ∈ inv(Succ[C ][L])} and to {s[X ← E(s)] : s ∈ inv(L)} ⊆ inv(Succ[C][L]))), If Step[C][L] is B then we have to check that : {final <s', L'> : s' ∈ {s} ∧ L' ∈ {Succ[C][L](s ∈ B)} } ⊆ [{<s", L"> : s" ∈ inv(L") ∧ L" ∈ Lab[C] {√}} ∪ inv(√)] ⇔ final <s, Succ[C][L](s ∈ B)> ∈ [{<s", L"> : s" ∈ inv(L") ∧ L" ∈ Lab[C] {√}} ∪ inv(√)] ⇔ s ∈ inv(Succ[C][L](s ∈ B)). • Finally, (γ(inv) ∩ S ⊆ q) ⇔ (inv(√) ⊆ q) [by (36.2)] ⇔ (∀ L ∈ After[C]. inv(L) ⊆ q) [by (32.7)]. 5.2 .4 Semantical soundness and completeness of the stepwise Floyd-Naur partial correctness proof method A proof method is sound if it cannot lead to mistaken conclusions. It is complete if it is always applicable to prove indubitable facts. THEOREM DE BAKKER & MEERTENS [1975] Soundness and semantical completeness (46) of the stepwise Floyd-Naur method The stepwise presentation of Floyd-Naur partial correctness proof method is semantically sound and complete. Proof The method is sound since if inv satisfies (45) then, by construction of (45), gvc[C][p, q]( γ(inv) ) holds so that { p }C{ q } derives from (29). It is semantically complete since if { p }C{ q } is true then by (29) we know that I = {<s, C> : s ∈ p}  op[C]* satisfies gvc[C][p,q](I) so that by construction, (45) holds for inv = α(I). We insists upon semantical soundness and completeness as in DE BAKKER & MEERTENS [1975] or MANNA & PNUELI [1970] since (46) is relative to a given semantics of programs (13) and to a representation of invariants by sets as opposed to the existence of a formal calculus in a given language to prove partial correctness of programs (GERGELY & SZÖTS [1978], SAIN [1985]). 5 .3 The compositional Floyd-Naur partial correctness proof method HOARE [1969] introduced the idea (often called compositionality) that the specification of a command should be verifiable in terms of the specifications of its components. This means that partial correctness should be proved by induction on the syntax of programs using their relational semantics (19) instead of an induction on the number of transitions using their operational semantics (13). Following OWICKI [1975], we give a syntax-directed presentation of Floyd-Naur's method without appeal to a formal logic. To do this we associate preconditions and postconditions with commands and introduce structural verification conditions so that a proof of a composite command is composed of the proofs of its constituent parts. Although this later turns out to be redundant, we prove the semantical soundness and completeness of the method since the underlying reasoning constitutes a simple introduction to relative completeness proofs of Hoare logic. 5.3 .1 Preconditions and postconditions of commands A partial correctness proof of { p }C{ q } by Floyd-Naur's method consists in discovering a precondition pre(C') and a postcondition post(C') specifying the partial correctness {pre(C')}C'{post(C')} of each component C' of command C. This includes an invariant linv(C') for each loop C' within C. Formally “pre”, “post” and “linv” can be unde

Methods and Logics for Proving Programs.

ion. What would be left if we abstracted from all contentful functionand objectsenses in a Fregean thought would be nothing but a pattern of functional applications and abstractions. For example, instead ofions. For example, instead of for every two integers, there is a third that is greater than their sum, i.e., (∀x)(∀y)((Ix & Iy)⊃(∃z)(Iz & z>x+y)), which we might rewrite in functional notation as ∀(λx∀(λy⊃(&(I(x),I(y)),∃(λz>(z,+(x,y)))))), we would have the “form” Φ(λxΦ(λyζ(θ(π(x),π(y)),Ψ(λzκ(z,μ(x,y)))))). But at this level of abstraction the logical structure of the claim is no longer fully in view, since we’ve “abstracted from” the logical constants. We could continue to maintain that logic is unrestrictedly formal, but only at the cost of confining logic to the meager resources of function application and abstraction, and removing conjunction, negation, quantification, and other notions generally regarded as logical from its purview. It seems more reasonable to follow Frege in denying that logic is unrestrictedly formal.25 For Frege, then, logical notions cannot be distinguished from non-logical ones on by generic features of their semantic role. Both the logical notion identity and the non-logical notion is taller than are first level functions with two arguments. They differ only in their graphs: One puzzle: in the long passage quoted from FG2, Frege lists subsumption (the falling of an object under a concept or a first level concept under a second level concept, PW:193, 213) as one of the notions proper to logic. But subsumption is not a concept or relation, but the notion of functional application itself (or rather a restriction of that notion, since not all functions are concepts). What is more, subsumption does not correspond to any symbol in the Begriffsschrift, so logical relations that depend on it would be preserved under arbitrary category-preserving translations. It is therefore hard to see how it belongs with identity, negation, and concept subordination in Frege’s list. CHAPTER 5. FREGE AND THE FORMALITY OF LOGIC 157 X Y X is taller than Y X is identical with Y George Bush Jesse Jackson False False Odysseus Penelope True False George Bush George Bush False True etc. etc. etc. etc. Similarly, both the universal quantifier and the second level concept some philosophers are second level functions with one argument. The line between logical and non-logical notions must therefore appeal to specific features of items within each semantic category (perhaps permutation invariance or something similar). For Kant, by contrast, there is a generic difference in semantic functioning between logical and non-logical vocabulary. Notice that Frege’s argument against the 3-formality of logic in FG2 does not depend on his commitment to logical objects and his logicist program (though surely it is motivated, at least in part, by these commitments). One could consistently hold that logic has its own contentful concepts (negation, identity, concept subordination, etc.) while denying that objects can be given through logical means alone. This observation will be important in section 5.4, when we ask where we should stand in the debate between Frege and Kant on the nature of logic, given that most philosophers now reject the idea that logic has its own objects. 5.2.6 “The most general laws of truth” To hold that logic is 1-formal but not 3-formal, as Frege does, is to reject Kant’s Thesis. On Frege’s view, the laws of logic are both constitutive norms for thought as such and informative about very general features of the “nature of things.” Frege devotes considerable attention (from 1893 on) to explaining how this can be so. Frege’s main concern is to defuse an argument that would establish Kant’s Thesis on quite general grounds (grounds independent of any specifically Kantian doctrines). The argument begins with the observation that “law” is used in two senses, which we might call “normative” and “descriptive:” “In one sense a law asserts what is; in the other it CHAPTER 5. FREGE AND THE FORMALITY OF LOGIC 158 prescribes what ought to be” (GGZ:xv). A normative law prescribes what one ought to do or provides a standard for the evaluation of one’s conduct as good or bad. Public statutes, moral codes, and aesthetic canons are all normative laws. A descriptive law, on the other hand, describes counterfactually robust regularities in the order of things. Laws of physics, biology, and geometry are descriptive laws. Whether a law is normative or descriptive is an intrinsic feature of its content: no law can be both. The argument Frege is concerned to rebut simply applies this exclusive categorization to the laws of logic. If these laws are normative for thought as such (1-formal), then they cannot be descriptive: they cannot say anything about how things are. Hence, if they are 1-formal, they must also be 3-formal. Against this line of thought, Frege wants to have it both ways: he wants to conceive logical laws both as prescribing how one ought to think and as saying how things are. He does not think that logical laws are explicitly prescriptive as to their content (Ricketts 1996:127). They have the form “such and such is the case,” not “one should think in such and such a way”: The word ‘law’ is used in two senses. When we speak of moral or civil laws we mean prescriptions, which ought to be obeyed but with which actual occurrences are not always in conformity. Laws of nature are general features of what happens in nature, and occurrences in nature are always in accordance with them. It is rather in this sense that I speak of laws of truth [i.e., laws of logic]. Here of course it is not a matter of what happens but of what is. (Th:58) But although logical laws are not prescriptive in their content, they imply prescriptions and are thus prescriptive in a broader sense: “From the laws of truth there follow prescriptions about asserting, thinking, judging, inferring” (Th:58, emphasis added). Hence Any law asserting what is, can be conceived as prescribing that one ought to think in conformity with it, and is thus in that sense a law of thought. This holds for laws of geometry and physics no less than for laws of logic. The latter have a special title to the name “laws of thought” only if we mean to assert that they are the most general laws, which prescribe universally the way in which one ought to think if one is to think at all. (GGZ:xv) Frege’s line of thought here is subtle enough to deserve a little unpacking. Consider the statement “the white King is at C3.” Though the statement is descriptive in its content, it CHAPTER 5. FREGE AND THE FORMALITY OF LOGIC 159 has prescriptive consequences in the context of a game of chess: for instance, it implies that white is prohibited from moving a bishop from C4 to D5 if there is a black rook at C5. Now instead of chess, consider the “game” of thinking about the physical world. As in chess, “moves” in this game—thoughts—can be assessed as correct or incorrect. Thoughts about the physical world are correct to the extent that they accurately depict the way the world is. Laws of physics (for instance, Maxwell’s equations) are descriptive laws; they tell us how the physical world is. But in the context of the “game” or activity of thinking about the physical world, they have prescriptive consequences: one ought not make judgments that conflict with them. In so far as one’s activity is to count as thinking about the physical world, it must be assessable as correct or incorrect (true or false) by reference to the laws of physics.26 In this sense, the laws of physics provide constitutive norms for the activity of thinking about the physical world. This is not to say that one cannot think wrongly about the physical world: one’s thoughts need not conform to the norms provided by the laws of physics; they need only be assessable in light of them. Nor is it to say that one must be aware of these laws in order to think about the physical world. The point is simply that to count someone as thinking about the physical world is ipso facto to hold her thoughts assessable for truth or falsity by their agreement or disagreement with the laws of physics. Someone whose thoughts were not so assessable could still be counted as thinking, but not as thinking about the physical world. It is in this sense that Frege holds that a law of physics “. . . can be conceived as prescribing that one ought to think in conformity with it, and is thus in that sense a law of thought.” On Frege’s view, then, laws of physics cannot be distinguished from laws of logic on the grounds that the former are descriptive and the latter prescriptive. Both kinds of laws are descriptive and have prescriptive consequences. They differ only in the activities for which they provide constitutive norms. While physical laws provide norms for thought about the physical world, logical laws provide norms for thought as such. To count an activity as There are of course other dimensions of correctness of thought besides truth and falsity: e.g., standards for proper justification. CHAPTER 5. FREGE AND THE FORMALITY OF LOGIC 160 thought about the physical world is to hold it assessable in light of the laws of physics; to count an activity as thought at all is to hold it assessable in light of the laws of logic. (As in the case of the laws of physics, there is no implication that thought must conform to these laws in order to count as thought: the point is that they must be assessable in light of these laws.)27 Frege often casts the difference between logical laws and laws of the special sciences as a difference in generality : logical laws are more general in the sense that they “. . . prescribe universally the way in which one ought to think if one is to think at all” (GGZ:xv), as opposed to the way in which one ought to think in some particular domain (cf. PW:1456). Thus, in his 1897 “Logic” manuscript, Frege writes: Like ethics, logic c

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What does it mean to say that logic is formal

The rapid growth of video data demands both effective and efficient video summarization methods so that users are empowered to quickly browse and comprehend a large amount of video content. It is a herculean task to manage access to video content in real time where humongous amount of audiovisual recorded data is generated every second. In this paper we propose an Eratosthenes Sieve based key-frame extraction approach for video summarization (VS) which can work better for real-time applications. Here, Eratosthenes Sieve is used to generate sets of all Prime number frames and nonprime number frames up to total N frames of a video. k-means clustering procedure is employed on these sets to extract the key–frames quickly. Here, the challenge is to find the optimal set of clusters, achieved by employing Davies-Bouldin Index (DBI). DBI a cluster validation technique which allows users with free parameter based VS approach to choose the desired number of key-frames without incurring additional computational costs. Moreover, our proposed approach includes likes of both local and global perspective videos. The method strongly enhances clustering procedure performance trough engagement of Eratosthenes Sieve. Qualitative and quantitative evaluation and complexity computation are done in order to compare the performances of the proposed model and state-of-the-art models. Experimental results on two benchmark datasets with various types of videos exhibit that the proposed methods outperform the state-of-the-art models on F-measure.

Eratosthenes sieve based key-frame extraction technique for event summarization in videos

The history of eighteenth-century philosophy is a subject with its own history. However, the idea of what constitutes eighteenth-century philosophy has been remarkably stable over the two centuries that have elapsed since the period in question, and this stability has obscured the simple fact of its historicity and made it peculiarly difficult to question the historical adequacy of that idea. What is more, even now, when detailed scholarship has undertaken such questioning in earnest, tradition is so strong that works of synthesis and overview – not to mention teaching – have to pay it considerable respect in order to find an identifiable audience. During the last two centuries, two factors above all have lent the philosophy of the eighteenth century an identity, other than its place in time, and these two factors have often reinforced each other. One is the idea that the philosophy in question is the core of a wider cultural and social movement, namely ‘the Enlightenment’. The other is that the eighteenth century has to be seen as part of – in fact, as the high point of – a development of early-modern philosophy from Francis Bacon and Rene Descartes to Thomas Reid and Immanuel Kant. THE ENLIGHTENMENT AND PHILOSOPHY The attempt to identify the philosophy of the eighteenth century by means of the Enlightenment is as inadequate as it is popular. Apart from the danger of tautology – namely that the philosophy of the eighteenth century is the philosophy of the Enlightenment because the Enlightenment is the eighteenth century – the concept of Enlightenment is either too wide or too narrow to capture the philosophical riches of the century.

The history of eighteenth-century philosophy: history or philosophy?

1. Frederick William I. of Prussia ordered that his son, known later as Frederick the Great, should "learn no Latin"; "let him learn arithmetic, mathematics, artillery,-economy to the very bottom."' The old king allowed the Berlin "Society of Sciences," the favorite child of Leibniz, to languish and almost to pass away. His son, Frederick, on the other hand, secretly acquired some Latin, shunned the study of mathematics beyond its rudiments, and brought the Berlin "Academy" 2 to great splendor. Frederick looked upon mathematical study with disfavor. As crown prince, he wrote, on January 26, 1738, to Voltaire his plan of study,3 "to take up again philosophy, history, poetry, music. As for mathematics, I confess to you that I dislike it; it dries up the mind. We Germans have it only too dry; it is a sterile field which must be cultivated and watered constantly, that it may produce." 2. Bantering the mathematicians. D'Alembert once wrote to Frederick the Great :4 "It is the destiny of your majesty to be always at war; in summer with the Austrians, in winter with mathematics." The king himself put it in these words :"I love to wrangle with mathematicians, that I may know whether, without understanding xx+y, it is not possible to be in the right." The king presents an argument why D'Alembert should not decline his invitation to settle in Berlin, ending with the remark :' "Such is my refutation. I hold myself to be victorious, and erect a trophy to myself, for having vanquished a great mathematician, wholly to his disgrace." Another time, he commentated on some essays of D'Alembert :7 "I read that part of the work in which you condescend to sink the science of the sublime geometry to the level of my ignorance."

Frederick the Great on Mathematics and Mathematicians

A prime number is a number other than 1 which has exactly two factors itself and 1. Eratosthenes discovered a method of finding prime numbers called the Sieve of Eratosthenes. Use Eratosthenes' algorithm (described below) on the chart on the opposite side of this sheet to find all the prime numbers between 1 and 100. Eratosthenes used a metal plate, and presented the plate to Ptolemy III, King of Egypt. Eratosthenes' algorithm: One is not a prime. Punch a small hole. Two is a prime number. Circle it. Two is prime. Since all multiples of 2 have themselves, 2, and 1 as factors, they are not prime. Punch a small hole for each. Three is the next number that is not crossed out. Circle it. Three is prime. Since all multiples of three are not prime, punch a small hole for each.

The sieve of eratosthenes.

(1928). The Early History of Partial Differential Equations and of Partial Differentiation and Integration. The American Mathematical Monthly: Vol. 35, No. 9, pp. 459-467.

Florian Cajori

Papers

Frederick the Great on Mathematics and Mathematicians

The sieve of eratosthenes.

The Early History of Partial Differential Equations and of Partial Differentiation and Integration

Generalizations in Geometry as Seen in the History of Developable Surfaces

Controversies on Mathematics between Wallis, Hobbes and Barrow