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Lorenzo Mascheroni 's 1797 work La geometria del compasso, which develops a geometry based solely on compass constructions, is considered by the author as stepping back behind the “demarcation line” of Euclidean geometry. In this work Mascheroni emphasizes the practical aspects of this geometry over a theoretical approach. A century later, in 1899, David Hilbert and his student Michael Feldblum proposed a totally different approach – algebraic and axiomatic – concerning geometric constructions based on various instruments. Taking into account that, at the end of the 18th century, straightedge geometry was also developed, one may ask what happened to the image of instrument-based geometry during the 19th century? By focusing on Mascheroni's book and its reception, this article aims to examine the various views and conceptions of mathematicians with respect to this geometry.

Given a set of sources and a set of sinks as points in the Euclidean plane, a directed network is a directed graph drawn in the plane with a directed path from each source to each sink. Such a network may contain nodes other than the given sources and sinks, called Steiner points. We characterize the local structure of the Steiner points in all shortest-length directed networks in the Euclidean plane. This characterization implies that these networks are constructible by straightedge and compass. Our results build on unpublished work of Alfaro, Campbell, Sher, and Soto from 1989 and 1990. Part of the proof is based on a new method that uses other norms in the plane. This approach gives more conceptual proofs of some of their results, and as a consequence, we also obtain results on shortest directed networks for these norms., 21 pages, 14 figures. To appear in Graphs and Combinatorics

Straightedge and compass constructions can he used to produce line segments of various lengths relative to some preassigned unit length. Although the lengths are all real numbers, it turns out that not every real number can be obtained in this way. The lengths which can be constructed are rather special.

In the previous chapters we developed the algebraic machinery for proving that the three famous geometric constructions are impossible. In this chapter we introduce some geometry and start to show the connection between algebra and the geometry of constructions.

Projective geometry can be described as the geometry of the straightedge, in comparison with Euclid’s geometry of straightedge and compass.

We examine how three authors, Dante Alighieri, Margaret Cavendish, and James Joyce, employed allusions to the classical problem of squaring the circle, i.e., using only a compass and a straightedge to construct a square whose area equals that of a given circle. These discussions will be against the background of what was known mathematically about solving the problem when they wrote—respectively, in the fourteenth, seventeenth, and twentieth centuries. We will also discuss, as best as can be determined, what each author believed, knew, or could have known about the mathematical status of the problem and what this reveals about their use of the construction in their writings.

The geometric constructions obtained with only straightedge and compass are famous and play a special role in the development of geometry. On the one hand, the constructibility of figures is a key ingredient in Euclid geometry and, on the other hand, unconstructibility gave birth to famous open problems of the ancient Greece which were unlocked only in the nineteenth century using discoveries in algebra. This paper discusses the mechanization of straightedge and compass constructions. It focuses on the algebraic approaches and presents two methods which are implemented; one is due to Lebesgue and the other one was jointly designed by Gao and Chou. Some links between the algebraic approach of constructions and synthetic geometry are described.

Constructions are central to the methodology of geometry presented in the Elements. This theory therefore poses a unique challenge to those concerned with the practice of constructive mathematics: can the Elements be faithfully captured in a modern constructive framework? In this paper, we outline our implementation of Euclidean geometry based on straightedge and compass constructions in the intuitionistic type theory of the Nuprl proof assistant. A result of our intuitionistic treatment of Euclidean geometry is a proof of the second proposition from Book I of the Elements in its full generality; a result that differs from other formally constructive accounts of Euclidean geometry. Our formalization of the straightedge and compass utilizes a predicate for orientation, which enables a concise and intuitive expression of Euclid’s constructions.

In this paper, I present an interpretation of the use of constructions in both the problems and theorems of Elements I–VI, in light of the concept of given as developed in the Data, that makes a distinction between the way that constructions are used in problems, problem-constructions, and the way that they are used in theorems and in the proofs of problems, proof-constructions. I begin by showing that the general structure of a problem is slightly different from that stated by Proclus in his commentary on the Elements. I then give a reading of all five postulates, Elem. I.post.1–5, in terms of the concept of given. This is followed by a detailed exhibition of the syntax of problem-constructions, which shows that these are not practical instructions for using a straightedge and compass, but rather demonstrations of the existence of an effective procedure for introducing geometric objects, which procedure is reducible to operations of the postulates but not directly stated in terms of the postulates. Finally, I argue that theorems and the proofs of problems employ a wider range of constructive and semi- and non-constructive assumptions that those made possible by problems.

Review of Punk Rock is My Religion: Straightedge Punk and 'Religious Identity, by Francis Stewart. Punk & Post-Punk, 7(2) (2018), pp. 271-274.

The accuracy of 5-axis machine tools is a key factor to modern manufacturing of multi-axes machined workpieces. Up to 75% of errors on manufactured workpieces are caused by thermally induced errors of machine tools. In this paper a thermal test piece for evaluating thermal errors of machine tools equipped with rotary tables is introduced. In comparison to geometric test pieces, which detect geometrical errors of machine tools only, the developed thermal test piece visualizes thermally induced errors in X-, Y- and Z-direction as well as two additional angular errors at the tool center point (TCP) of 5-axis machine tools over time. Further, the thermal material expansion error of the test piece on centric clamped machined workpieces can be evaluated with the new thermal test piece. The presented thermal test piece is machined during a corresponding eight hour test cycle and tested on two 5-axis milling machines with different axis configurations. The thermal test piece can either be evaluated with a coordinate measuring machine (CMM) or manually with handheld measurement devices such as dial gauge, micrometer and straightedge. The thermal behavior of the two machine tools is also investigated, following the ISO 230-3:2007 regarding thermal errors of machine tools, under no-load or finishing conditions using the R-Test measurement setup. The resulting workpiece errors evaluated by using the thermal test piece are compared to the machine tool measurements. A good correspondence between the R-Test measurements and the thermal workpiece errors can be shown. CMM and handheld measurements are both capable to detect the thermal machine tool errors.

Summary. The question of performing, by straightedge and compass only, the classic Greek construction problem of duplicating a cube remained a significant open problem in mathematics until the firs...

We present a collection of simple construction problems and determine conditions for which the construction can be carried out with compass and straightedge. Since the problems reduce to finding th...

Despite the frequency and intensity of scholarly attention to Plato's important divided line passage in Republic 6, an ancient puzzle has repeatedly baffled interpreters: the two middle segments of the line are of equal length, although Socrates declares that the degree of clarity and truth increases in proportion to the contents of each of the line's four segments. The missing piece of the puzzle, I argue, is provided by the practice of ancient Greek geometry: diagrams were drawn with a compass and straightedge. Thus, as actually constructed in the fifth and fourth centuries BCE, the diagram itself vindicates Plato's text.

The first three books of Euclid’s Elements provide the charming classical introduction to deductive mathematics with the geometry of the blackboard. A blackboard with straightedge and a pair of com...

In order to promote geometric understanding, teachers frequently use hands-on activities. Such activities can be used to expound upon the declarative statements and theorems of geometry. Using a compass, straightedge, and protractor, students are able to actively build conceptions involving bisectors, midpoints, and perpendicular lines. Additionally, activities that require students to problem-solve and formulate problems, using their construction knowledge and skills, can reinforce and strengthen that which they have learned. This article describes STEAM instruction with high school geometry students designed to productively integrate geometric constructions, digital technology, elements of art, and principles of design to enhance students’ geometric reasoning.

This paper is devoted to exposition of a provable classical solution for the ancient Greeks classical geometric problem of angle trisection [3]. (Pierre Laurent Wantzel, 1837),presented an algebraic proof based on ideas from Galois field showing that, the angle trisection solution correspond to an implicit solution of the cubic equation; , which he stated as geometrically irreducible [23]. The primary objective of this novel work is to show the possibility to solve the trisection of an arbitrary angle using the traditional Greeks tools of geometry, and refutethe presented proof of angle trisection impossibility statement. The exposedproof of the solution is theorem , which is based on the classical rules of Euclidean geometry, contrary to the Archimedes proposition of usinga marked straightedge construction [4], [11].

This paper presents a precision measurement of the Z-slide vertical error motion of an ultra-precision lathe based on a three-probe measurement method. The vertical error motion of the Z-slide, which is composed of the vertical parallelism error motion and the vertical straightness error motion, is measured by utilizing three displacement measuring probes to scan a self-cut roll workpiece and an accurate straightedge, respectively. A series of factors which could influence the measurement of the vertical error motion, including the error motion of the spindle, the surface form error of the reference object and the horizontal error motion of the Z-slide, are all successfully removed by using the proposed three-probe measurement system with an error separation method. The influence of the gravity induced deflection of the roll workpiece on the measurement of the Z-slide vertical parallelism error motion is also discussed. Experiments were carried out to verify the feasibility of the proposed methods on a three-axis ultra-precision lathe. The vertical straightness error motion was evaluated to be 133 nm with a measurement repeatability of 21 nm and the vertical parallelism error motion was evaluated to be 12.1 μm with a measurement repeatability of 38 nm.

This book grew from the idea that much of projective geometry is the elaboration of a simple concept, the central collineation. A central collineation is a construction, carried out with just a straightedge and a device to construct parallel lines, following the simplest rules, that transforms one diagram into another. We call it a collineation because a line is always transformed to a line. We’ll soon explain the meaning of central. And how is a circle transformed? That is the marvelous part, for a circle becomes a parabola or ellipse or hyperbola—a conic section. And all conic sections can be formed this way.

We discuss the basic fold rules of origami. With a small number of fold rules, we can construct geometric shapes that we see at geometry classes of schools. We construct those shapes usually by a straightedge and a compass, so-called a Euclidian tool of construction. We explain the set of the basic fold rules and show, by examples, that it is as powerful as a straightedge and a compass. Furthermore, we show that the set of basic fold rules enables us to construct the shapes by folding by hand. The set of the basic fold rules is the main ingredient of more powerful Huzita-Justin’s fold rules that we discuss in Chapter 3.

We discuss an origami geometry based on Huzita-Justin’s basic folds. In the 2D Euclidean geometry, we use a straightedge and a compass to construct geometric objects. We show that Huzita-Justin’s basic folds can construct them without such tools but by hand. We reformulate Huzita-Justin’s fold rules by giving them precise conditions for their use. We prove that we can decide whether, by the reformulated rules, we can perform a fold as specified by the parameters. Furthermore, we show that the rules yield a finite number of fold-lines. We present an alternative single general fold based on one of Huzita-Justine’s fold rules without incident and degenerate conditions. We show how the general fold rule can realize each of the reformulated folds. Huzita-Justin’s fold rules are more powerful than a straightedge and a compass as a tool of geometric construction. The famous examples to verify this fact are computing a cube root of a positive number and trisecting an arbitrary angle. We show the concrete programs to solve the problems.

Roughly, a real number a is called a constructible number if by the application of straightedge and compass we can construct, given a line segment of unit length, a line segment of length a. In some familiar geometric situations, we shall apply the results of Galois theory. In our general development, we shall show that the general polynomial equation of degree five has no solution in radicals.

The Texas Department of Transportation (DOT) ride standard specifies two devices for measuring ride quality on pavement projects: the 10-ft (3-m) straightedge and the inertial profiler. According to this specification, inertial profilers measure the ride quality on projects longer than 2,500 ft (762 m). The 10-ft straightedge must be used to survey roughness for projects of less than 2,500 ft. A study was conducted to identify the potential issues associated with the use of inertial profilers and the international roughness index (IRI) algorithm on short projects. A literature review was performed to collect required information about the inertial profiler operation and the IRI algorithm. A field experiment was conducted with an inertial profiler certified by the Texas DOT. The collected data were analyzed to investigate the effect of profiling speed and segment length. The results indicate that inertial profilers can be used to collect profile data on short segments of less than 2,500 ft, but the IRI algorithm should be used on projects longer than 528 ft (161 m), below which the variation between IRI results increases significantly.

This paper examined the sensitivity of 20 road unevenness indicators (RUIs) for local discontinuities (various distresses, joints, joint and surface defects, cracks, spalls, other road features, humps, dips, etc.) of a maximum length of 20 cm in Portland cement concrete pavements. Approximately, 5500 road records totalling 530 km from the Long-Term Pavement Performance programme were processed. Raw profiles were separated into random and distress components using a median filtering method. The median filter was set to identify distress of a maximum length of 20 cm and with a minimal height of 3 mm. Over 26,000 distresses and their dimensions were identified. Raw longitudinal road profiles were compared with the separated pure random components. The most sensitive indicators of distresses were road elevation spectrum parameters in whole- and short-wave bands, the Profile Index and the road elevation root mean square (RMS) value in the short-wave band. Negligible sensitivity was observed for the Int...

Straightedge and compass constructions play a special role in geometry. First, for a very long time, they were used in practice by land surveyors or architects in order to solve concrete problems. Second, they are an inexhaustible source of exercises used to learn concepts in geometry. And finally, some famous impossible problems related to straightedge and compass constructions waited centuries before being solved using algebra. In addition, not knowing if a well-constrained problem is constructible with straightedge and compass or not, make that kind of problems more difficult to address: should we synthesize a program or on the contrary find a counterexample has to be found and treated using algebra or reduction on it? In this paper, we perform a systematic checking of a whole corpus of problems proposed by William Wernick. We expose our methodology and the algorithm we used. For each problem, its constructibility status is computed and either an algebraic argument or a geometric construction is given.

Straightedge-and-compass construction problems are well known for different reasons. One of them is the difficulty to prove that a problem is not constructible: it took about two millennia to prove that it is not possible in general to cut an angle into three equal parts by using only straightedge and compass. Today, such proofs rely on algebraic tools difficult to apprehend by high school student. On the other hand, the technique of problem reduction is often used in theory of computation to prove other kinds of impossibility. In this paper, we adapt the notion of reduction to geometric constructions in order to have geometric proofs for unconstructibility based on a set of problems known to be unconstructible. Geometric reductions can also be used with constructible problems: in this case, besides having constructibility, the reduction also yields a construction. To make the things concrete, we focus this study to a corpus of triangle location problems proposed by William Wernick in the eighties.

To indicate the inability to achieve an enterprise or the futility of some theoretical disquisitions, a range of idiomatic expressions are frequently used in our daily conversation, such as discussing “how many angels can stand on the point of a pin” This saying comes from the long disputes of medieval scholars about angelology, which now appear of modest if not totally irrelevant interest. Another of these expressions, “squaring the circle,” is used as a metaphor for an impossible task, for example, the search for perpetual motion which kept busy so many inventors for centuries. For quite a long time, some of the brightest minds dedicated intense efforts to the attempt to square the circle, i.e., to the construction of a square with the same area as a circle, using only a compass and straightedge and a finite number of operations. This seemingly simple problem challenged generations of mathematicians, professional, as well as amateur, until in 1882 the impossibility of such a construction was finally proven.

The rose window on the main facade of Orvieto cathedral is unique because its design is based on a 22-sided polygon. It is well known that the icosikaidigon cannot be constructed using only a compass and a straightedge. Therefore, using a geometric/statistical approach, we intend to unveil which approximate construction method was used by the sculptor to draw this unique rose window.

This article uses forty-four face-to-face interviews with individuals who formerly identified with straightedge—a clean-living, mostly youth-based (sub)culture—to explore the possible role chosen youth cultural identities play in adult transition, as well as extend recent work on aging and youth scenes by more deeply engaging both “subjective adulthood” and the retrospective accounts of “ex” members. Data show interviewees developing (paths to) subjective adulthoods substantially influenced by former affiliation with straightedge culture they frequently believe mark their (paths to) adulthoods fundamentally distinct from others in their age cohort. Particularly, individuals transitioning from straightedge recounted pronounced subculturally rooted antipathy toward adult conventionality, often envisioned alternative adult trajectories for themselves, discussed transitional impediments and opportunities they took to be unique to transitioning from straightedge, and, in indicating heightened awareness of adulthood's “facework,” visualized a collective of others like them inside adult social spheres by virtue of the formative bases (former) scene affiliation provided them. Ultimately, findings suggest that the study of subjective adult transition may profit from directly considering the formative influence of elective youth identities. Likewise, perhaps the most fertile grounds in the turn toward examining aging and scenes might rest in meanings individuals attach to adulthood and transitioning, even for ex-members of certain communities.

This article explores the formation of Krishnacore, a phenomenon born from the amalgamation of American straightedge punk and the Hare Krishna Movement in the 1990s. It argues that whilst shared choices of lifestyle, such as vegetarianism and a distaste for intoxicants and illicit sex, were core tenets towards the conception of the scene, it was bhakti-yoga (the theological and philosophical basis of the Hare Krishna Movement) that cemented such a relationship. Furthermore, it also explores the aesthetic context of punk within a Vedic context, in particular with reference to what is termed as ‘Nada-Brahma’, or the sacralisation of sound.

In many areas, some geometry problems can not be solved using only geometry and are treated by the means of algebraic tools. However, geometric properties can be still employed to simplify the system of equations. This allows either to speed up the treatment or, more radically, to make the treatment possible. In this article we illustrate this approach with a family of toy examples. In all these problems the goal it is to determine if there is a compass-and-straightedge construction of the three vertices of a triangle knowing only three located points of this triangle. Algebraic tools, basically Galois theory, are needed to answer the question. But in many cases a geometric reasoning phase is required to provide a polynomial algebraic system that algebraic softwares can address within a acceptable time despite the exponential complexity of the underlying algorithms.

According to some current problems in the course of measuring the plane shape error of workpiece, an in-situ measuring method based on laser triangulation is presented in this paper. The method avoids the inefficiency of traditional methods like knife straightedge as well as the time and cost requirements of coordinate measuring machine(CMM). A laser-based measuring head is designed and installed on the spindle of a numerical control(NC) machine. The measuring head moves in the path planning to measure measuring points. The spatial coordinates of the measuring points are obtained by the combination of the laser triangulation displacement sensor and the coordinate system of the NC machine, which could make the indicators of measurement come true. The method to evaluate planar straightness error adopts particle swarm optimization(PSO). To verify the feasibility and accuracy of the measuring method, simulation experiments were implemented with a CMM. Comparing the measurement results of measuring head with the corresponding measured values obtained by composite measuring machine, it is verified that the method can realize high-precise and automatic measurement of the planar straightness error of the workpiece.

In mathematical problem-solving, the emphasis is often on the classroom use of so-called open problems. According to some, problem-solving is best stimulated by providing open-ended mathematical tasks. Not only that, it is also argued that openness of problems is more conducive to students’ mathematical creativity compared to using closed tasks. In this chapter we examine this assumption and make a case for ‘constraints-based’ task design. In this approach, which has its roots in economic research on scarcity (as well as being exemplified by aspects of the American television series MacGyver), we argue that tasks that are ‘moderately closed’ (neither fully ‘open’ nor fully ‘closed’) can facilitate creative mathematical thinking and problem-solving. Using examples from a range of topics, we explore cases of ‘constraints-based’ creativity such as producing geometry constructions solely with straightedge and compass, ways of tackling number puzzles and solutions to sets of equations. We argue that such examples demonstrate that mathematical problem-solving and creativity need not solely necessitate open-ended mathematical problems rather that tasks with suitable constraints can serve as creativity-inducing problem-solving tasks as well.

We now apply our knowledge of field extensions in order to answer three questions posed by the ancient Greeks.

The final chapter is, of course, not a final conclusion concerning the “fruition” of the mathematics of paper folding: this domain is in a process of continuous development, and its interweaving with computer sciences and computer modeling becomes more and more apparent and prominent. This, of course, raises the question as to how materiality, which earlier functioned partially as a hindering element, is considered; one may wonder, for example, as to the epistemological implications and differences between the twenty-first century mathematical computer models and the nineteenth century mathematical material folded models, but this discussion is beyond the scope of this chapter. What this chapter will attempt to describe is that which can be called the first step of this new, modern movement concerning mathematics and folding-based geometry: the complete axiomatization of this geometry. Though not the single modern point of view with which the mathematics of paper folding has been considered, the attempt to find axioms or basic operations for this geometry, started at the beginning of the twentieth century, bore fruit only towards the end of the 1980s. If Beloch’s interest lay in proving that folding-based geometry is at least as powerful as straightedge and compass geometry, from an algebraic point of view as well as construction-wise, during the mid-1980s, one was interested in finding the basic, fundamental operations from which a folded figure was composed, and by this means, giving this geometry a logical sound basis, the same that straightedge and compass geometry had for centuries.

International audience; Geometric constructions have been a topic of interest among mathematicians for centuries. The Euclidean tradition uses two instruments: a straightedge (ruler) and a compass. However, limited constructions have also been studied. The Mohr–Mascheroni theorem says that every construction using both instruments can be replaced by a construction of the same object that uses only a compass. (Obviously, a line cannot be drawn with a compass; we construct two distinct points on that line instead.) Another result, the Poncelet–Steiner theorem, says that in a construction using straightedge and compass, a single application of a compass suffices: if a circle with its center is given, then every straightedge–compass construction can be performed with only a straightedge. Today, such results are considered to belong to recreational mathematics, and they can be found in many books, such as [6], for example.

The geometric process followed by Leon Battista Alberti when designing the facade of the Santa Maria Novella church in Florence is well-known. This facade contains 48 ornamental elements which were created through the construction of regular polygons. Specifically: 7 elements have a pentagonal base, 3 elements have an hexagonal base, 36 elements have an octagonal base, and 2 elements have an icosikaihexagonal base (26 sides). In our view, it is interesting that Alberti, having designed all ornaments on the facade on the basis of regular polygons which can be constructed using a straightedge and a compass only, decided to top the lateral scrolls with a circular design arising from a 26-sided regular polygon, since this regular polygon cannot be constructed using only a compass and a straightedge. Therefore, in this paper we use a mathematical approach to theoretically compare several approximate methods for constructing an icosikaihexagon using a compass and a straightedge, in order to ascertain which of these methods best suits the point pattern of this special Renaissance ornament.

We examine straightedge and compass constructions in spherical geometry. We show via examples that the starting conditions affect the set of constructible points. Although current tools do not allo...

We consider the problem of finding integer-sided triangles for which all three angles in the triangle can be trisected with a compass and unmarked straightedge. Since some angles (such as 60°) cann...

This study uses a new resin cement as a sealer material for manhole frames. This material is used for rapid manhole vertical adjustment (RMVA) techniques to complete a ‘Smooth Roads Project’. Precast screws within the newly installed manhole frame are used to adjust the elevation of the cover to match the road elevation and ensure a smooth transition from the road surface to cover. Using this technique, rapid vertical adjustment and replacement of the manhole is achieved. To evaluate the performance of the new resin cement used in RMVA techniques, a 3-meter straightedge was used to measure the smoothness between the manhole cover and the surrounding road surface. The smoothness tests have been carried out for 3·5 years to ensure a smooth ride after opening to traffic. Test results show that the measured values all fit in with the acceptance standard ±6 mm.

We discuss a new possible construction of the regular heptagon by rhombic bicompasses explained in the text as a new geometric mean of constructions in the spirit of classical constructions in connection with an unmarked ruler (straightedge). It avoids the disadvantages of the neusis construction which requires the trisection of an angle and which is not possible in classical way by compasses and ruler. The rhombic bicompasses allow to draw at once two circles around two fixed points in such correlated way that the position of one of the rotating points (arms) on one circle determines the position of the points on the other circle. This means that the positions of all points (arms) on both circles are determined in unique way.

For many centuries since ancient times the angle trisection has remained one of the three geometric problems that could not have a clear solution by straightedge and compass. Should such a solution had been, no one would have spent time on it. Moreover, no need would have been to prove that its solution is impossible due to the third degree equation. No need would have been to prove that it is impossible to construct a great number of regular polygons which now by this solution is absolutely possible. Furthermore, this solution by putting an end to all this, should also make us wonder whether Algebra can substitute Geometry especially when in an equation exist arithmetic factors behind which, a possibly different structure of it, is well hidden.

The study quantified the influence of the road profile obstacles on 27 different road unevenness indicators (RUI). The influence of obstacle height, length and number of obstacles per kilometre was assessed. The 2997 real longitudinal road-profile records from the Long-Term Pavement Performance program with considering asphalt concrete and cement concrete road surfaces were processed. 5169 obstacles were identified by a median filtering approach and their dimensions were quantified. The regression function based on linear coefficients for dependence of the RUI on three obstacle parameters was derived. The marked dependence of the road indicators on all three obstacle parameters was observed. The road indices describing the short wavelength band proved to be the most sensitive for detecting obstacle presence in a road profile.

In this study, effect of the sensor gain error is theoretically analyzed and simulated when mixed sequential two-prove method(MTPM) is applied for the precision measurement of straightness error of a linear motion table. According to the theoretical analysis, difference of the gain errors between two displacement sensors increases measurement error dramatically and alignment error of the straightedge is also amplified by the sensor gain difference. On the other hand, if the gain errors of the two sensors are identical, most of error terms are cancelled out and the alignment error doesn’t give any influence on the measurement error. Also the measurement error of the straightness error is minimized compared with that of the straightedge’s form error owing to close relationship between straightness error and angular motion error of the table in the error terms.

Among the regular polygons, the regular heptagon most certainly is Cinderella. Contrary to a triangle, a square, a pentagon or a hexagon, the heptagon is not constructible using compass and straightedge methods. In what follows, we will guide you through the proof of this non-constructibility.

According to legend, the island of Delos was hit by the plague. The local residents made offerings to the local oracle and asked what they had to do to stop the epidemic. The oracle answered that they should build a new cubic altar, with a size (= volume) double that of the existing one. The length of the edge should be determined by compass and straightedge methods only.

The earliest surviving trigonometric tables used to compute numerical values for geometrical magnitudes occur in Ptolemy’s Almagest (composed in the 2nd century AD). Current historical evidence cannot fix with precision the exact origins of such trigonometric tables. However, it seems likely that the Greek astronomer Hipparchos of Nicaea (2nd Century BC) was among the first, if not the very first to compute the ratio of chord to radius for a series of central angles in a circle, and to set the example of their use in astronomy for Ptolemy’s later work. By comparison, geometrical methods for the determination of magnitudes are considerably older, and have become highly formalized no later than the end of the 4th century BC, in Euclid’s Elements. This raises questions with regard to the comparative advantages of trigonometry over the older geometrical methods, and the particular emphasis that they received in the context of Greek mathematical astronomy.