Finsler geometry on the tangent bundle appears to be applicable to relativistic field theory, particularly, unified field theories. The physical motivation for Finsler structure is conveniently developed by the use of “gauge” transformations on the tangent space. In this context a remarkable correspondence of metrics, connections, and curvatures to, respectively, gauge potentials, fields, and energy-momentum emerges. Specific relativistic electromagnetic metrics such as Randers, Beil, and Weyl can be compared.

The analysis of a previous paper (see Ref. 1), in which the possibility of a Finslerian generalization of the equations of motion of gravitational field sources was demonstrated, is extended by developing the Finslerian generalization of the gravitational field equations on the basis of the complete contractionK = K lj lj of the Finslerian curvature tensorK l j hk (x, y). The relevant Lagrangian is constructed by the replacement of the directional variabley i inK by a vector fieldy i (x), (...) so that the notion of osculation may be regarded as the key concept on which the approach is based. The introduction of the auxiliary vector fieldy i (x) is shown to be of physical significance, for the field equations refer not only to the proper field variables but also to a special coordinate system associated withy i (x) through the Clebsch representation of the latter. The status of the conservation laws proves to be similar to that in the theory of the Yang-Mills field. By choosing a special Finslerian metric function we elucidate in detail the structure of the field equations in the static case. (shrink)

The issue of whether teleparallel nonlinear connections exist is resolved by their explicit construction on Finslerian metrics that arise in the Robertson test theory of special relativity (RTTSR), and on the Minkowski metric in particular. The method is an adaptation to the Finsler bundle of a similar construction for teleparallel linear connections. It suggests the existence of a concept of metric compatibility alternative toω μλ +ω λμ = 0 for teleparallel nonlinear connections. A sophisticated system of partial differential equations (...) whose solutions have been discussed in the computing literature is interpreted in geometric terms. The characteristics of the solutions are checked against compliance with the conditionω μλ +ω λμ = 0, an issue whose relevance for this theory derives from the fact that nonantisymmetric connections repeatedly appear in teleparallel geometry. (shrink)

Superluminal particles are studied within the framework of the Extended Relativity theory in Clifford spaces (C-spaces). In the simplest scenario, it is found that it is the contribution of the Clifford scalar component π of the poly-vector-valued momentum which is responsible for the superluminal behavior in ordinary spacetime due to the fact that the effective mass $\mathcal{M} = \sqrt{ M^{2} - \pi^{2} }$ is imaginary (tachyonic). However, from the point of view of C-space, there is no superluminal (tachyonic) behavior because (...) the true physical mass still obeys M 2>0. Therefore, there are no violations of the Clifford-extended Lorentz invariance and the extended Relativity principle in C-spaces. It is also explained why the charged muons (leptons) are subluminal while its chargeless neutrinos may admit superluminal propagation. A Born’s Reciprocal Relativity theory in Phase Spaces leads to modified dispersion relations involving both coordinates and momenta, and whose truncations furnish Lorentz-violating dispersion relations which appear in Finsler Geometry, rainbow-metrics models and Double (deformed) Special Relativity. These models also admit superluminal particles. A numerical analysis based on the recent OPERA experimental findings on alleged superluminal muon neutrinos is made. For the average muon neutrino energy of 17 GeV, we find a value for the magnitude $|\mathcal{M } | = 119.7\mbox{~MeV}$ that, coincidentally, is close to the mass of the muon m μ =105.7 MeV. (shrink)

The main features of how to build a Born’s Reciprocal Gravitational theory in curved phase-spaces are developed. By recurring to the nonlinear connection formalism of Finsler geometry a generalized gravitational action in the 8D cotangent space (curved phase space) can be constructed involving sums of 5 distinct types of torsion squared terms and 2 distinct curvature scalars ${\mathcal{R}}, {\mathcal{S}}$ which are associated with the curvature in the horizontal and vertical spaces, respectively. A Kaluza-Klein-like approach to the construction of (...) the curvature of the 8D cotangent space and based on the (torsionless) Levi-Civita connection is provided that yields the observed value of the cosmological constant and the Brans-Dicke-Jordan Gravity action in 4D as two special cases. It is found that the geometry of the momentum space can be linked to the observed value of the cosmological constant when the curvature in $\mathit{momentum}$ space is very large, namely the small size of P is of the order of $( 1/R_{\mathit{Hubble}})$ . Finally we develop a Born’s reciprocal complex gravitational theory as a local gauge theory in 8D of the $\mathit{deformed}$ Quaplectic group that is given by the semi-direct product of U(1,3) with the $\mathit{deformed}$ (noncommutative) Weyl-Heisenberg group involving four $\mathit{noncommutative}$ coordinates and momenta. The metric is complex with symmetric real components and antisymmetric imaginary ones. An action in 8D involving 2 curvature scalars and torsion squared terms is presented. (shrink)

I describe two approaches to modelling the universe, the one having its origin in topos theory and differential geometry, the other in set theory. The first is synthetic differential geometry. Traditionally, there have been two methods of deriving the theorems of geometry: the analytic and the synthetic. While the analytical method is based on the introduction of numerical coordinates, and so on the theory of real numbers, the idea behind the synthetic approach is to furnish the subject (...) of geometry with a purely geometric foundation in which the theorems are then deduced by purely logical means from an initial body of postulates. The most familiar examples of the synthetic geometry are classical Euclidean geometry and the synthetic projective geometry introduced by Desargues in the 17th century and revived and developed by Carnot, Poncelet, Steiner and others during the 19th century. The power of analytic geometry derives very largely from the fact that it permits the methods of the calculus, and, more generally, of mathematical analysis, to be introduced into geometry, leading in particular to differential geometry (a term, by the way, introduced in 1894 by the Italian geometer Luigi Bianchi). That being the case, the idea of a “synthetic” differential geometry seems elusive: how can differential geometry be placed on a “purely geometric” or “axiomatic” foundation when the apparatus of the calculus seems inextricably involved? To my knowledge there have been two attempts to develop a synthetic differential geometry. The first was initiated by Herbert Busemann in the 1940s, building on earlier work of Paul Finsler. Here the idea was to build a differential geometry that, in its author’s words, “requires no derivatives”: the basic objects in Busemann’s approach are not differentiable manifolds, but metric spaces of a certain type in which the notion of a geodesic can be defined in an intrinsic manner. I shall not have anything more to say about this approach. The second approach, that with which I shall be concerned here, was originally proposed in the 1960s by F.. (shrink)

We construct the Extended Relativity Theory in Born-Clifford-Phase spaces with an upper R and lower length λ scales (infrared/ultraviolet cutoff). The invariance symmetry leads naturally to the real Clifford algebra Cl (2, 6, R) and complexified Clifford Cl C (4) algebra related to Twistors. A unified theory of all Noncommutative branes in Clifford-spaces is developed based on the Moyal-Yang star product deformation quantization whose deformation parameter involves the lower/upper scale $$(\hbar \lambda / R)$$. Previous work led us to show from (...) first principles why the observed value of the vacuum energy density (cosmological constant) is given by a geometric mean relationship $$\rho \sim L_{\rm Planck}^{-2}R^{-2} = L_{P}^{-4} (L_{\rm Planck} / R)^{2} \sim 10^{-122}M_{\rm Planck}^4$$, and can be obtained when the infrared scale R is set to be of the order of the present value of the Hubble radius. We proceed with an extensive review of Smith’s 8D model based on the Clifford algebra Cl (1, 7) that reproduces at low energies the physics of the Standard Model and Gravity, including the derivation of all the coupling constants, particle masses, mixing angles,....with high precision. Geometric actions are presented like the Clifford-Space extension of Maxwell’s Electrodynamics, and Brandt’s action related to the 8D spacetime tangent-bundle involving coordinates and velocities (Finsler geometries). Finally we outline the reasons why a Clifford-Space Geometric Unification of all forces is a very reasonable avenue to consider and propose an Einstein-Hilbert type action in Clifford-Phase spaces (associated with the 8D Phase space) as a Unified Field theory action candidate that should reproduce the physics of the Standard Model plus Gravity in the low energy limit. (shrink)

The geometry of the Sagnac effect in a stationary region of a spacetime is reviewed with the aim of emphasizing the role of asymmetry of a Finsler metric defined on a spacelike hypersurface associated to a stationary splitting and related to future-pointing null geodesics of the spacetime. We show also that an analogous asymmetry comes into play in the Sagnac effect for timelike geodesics.

Traditionally, there have been two methods of deriving the theorems of geometry: the analytic and the synthetic. While the analytical method is based on the introduction of numerical coordinates, and so on the theory of real numbers, the idea behind the synthetic approach is to furnish the subject of geometry with a purely geometric foundation in which the theorems are then deduced by purely logical means from an initial body of postulates. The most familiar examples of the synthetic (...) geometry are classical Euclidean geometry and the synthetic projective geometry introduced by Desargues in the 17th century and revived and developed by Carnot, Poncelet, Steiner and others during the 19th century. The power of analytic geometry derives very largely from the fact that it permits the methods of the calculus, and, more generally, of mathematical analysis, to be introduced into geometry, leading in particular to differential geometry (a term, by the way, introduced in 1894 by the Italian geometer Luigi Bianchi). That being the case, the idea of a “synthetic” differential geometry seems elusive: how can differential geometry be placed on a “purely geometric” or “axiomatic” foundation when the apparatus of the calculus seems inextricably involved? To my knowledge there have been two attempts to develop a synthetic differential geometry. The first was initiated by Herbert Busemann in the 1940s, building on earlier work of Paul Finsler. Here the idea was to build a differential geometry that, in its author’s words, “requires no derivatives”: the basic objects in Busemann’s approach are not differentiable manifolds, but metric spaces of a certain type in which the notion of a geodesic can be defined in an intrinsic manner. I shall not have anything more to say about this approach. The second approach, that with which I shall be concerned here, was originally proposed in the 1960s by F. W. Lawvere, who was in fact striving to fashion a decisive axiomatic framework for continuum mechanics.. (shrink)

A connection viewed from the perspective of integration has the Bianchi identities as constraints. It is shown that the removal of these constraints admits a natural solution on manifolds endowed with a metric and teleparallelism. In the process, the equations of structure and the Bianchi identities take standard forms of field equations and conservation laws.The Levi-Civita (part of the) connection ends up as the potential for the gravity sector, where the source is geometric and tensorial and contains an explicit gravitational (...) contribution.Nonlinear field equations for the torsion result. In a “low-energy” approximation (linearity andlow energy-momentumtransfer), the postulate that only charge and velocities contribute to the source transforms these equations into the Maxwell system. Moreover, the affine geodesics become the equations of motion of special relativity with Lorentz force in the same approximation [J. G. Vargas,Found. Phys. 21, 379 (1991)]. The field equations for the torsion must then be viewed as applying to an electromagnetic/strong interaction.A classical unified theory thus arises where the underlying geometry confers their contrasting characters to Maxwell-Lorentz electrodynamics and to an Einstein's-like theory of gravity. The highly compact field equations must, however, be developed in phase-spacetime, since the connection is velocity-dependent, i.e., Finsler-like.Further opportunities for similarities with present-day physics are discussed: (a) teleparallelism allows for the formulation of the torsion sector of the theory as a flat space theory with concomitant point-dependent transformations; (b) spinors should replace Lorentz frames in their role as the subjects to which the connection refers; (c) the Dirac equation consistent with the frame bundle for a velocity-dependent metric with Lorentz signature generates a weak-like interaction in the torsion sector. (shrink)

Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the (...) aforementioned areas. (shrink)

§30. Significance of Desargues's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 CHAPTER VI. PASCAL'S THEOREM. §31. ...

The action-reaction principle (AR) is examined in three contexts: (1) the inertial-gravitational interaction between a particle and space-time geometry, (2) protective observation of an extended wave function of a single particle, and (3) the causal-stochastic or Bohm interpretation of quantum mechanics. A new criterion of reality is formulated using the AR principle. This criterion implies that the wave function of a single particle is real and justifies in the Bohm interpretation the dual ontology of the particle and its associated (...) wave function. But it is concluded that the Bohm theory is not dynamically complete because the particle and its associated wave function do not satisfy the AR principle. (shrink)

This paper aims to deal with the disputes on transferring demonstration between the various sciences in the context of the medicine-geometry relationship. According to Aristotle’s metabasis-prohibition, these two sciences should be located in separate compartments due to the characteristics of their subject-matter. However, a thorough analysis of the critical passage in Aristotle’s Posterior Analytics on circular wounds forces a revision of the boundaries of the interactions between sciences, since subsequently Avicenna, on the grounds of this passage, would widen the (...) area of the transference of demonstration. Furthermore, the fact that Avicenna and Ibn al-Nafīs continued to use geometrical demonstrations in their anatomical investigations shows the need to understand kind-crossing prohibition as a reminder to take into account the present scientific infrastructure and logical rules before proceeding onto a scientific investigation instead of accepting it as a mere nominal doctrine. Therefore, whether kind-crossing was possible or not depended on the extent to which the conclusion derived at the end of the scientific investigation, using a different method after taking into account all these reminders, had contributed to the solution of a particular proposition or the achievement of an approximate truth. (shrink)

There is little agreement about Aristotle’s philosophy of geometry, partly due to the textual evidence and partly part to disagreement over what constitutes a plausible view. I keep separate the questions ‘What is Aristotle’s philosophy of geometry?’ and ‘Is Aristotle right?’, and consider the textual evidence in the context of Greek geometrical practice, and show that, for Aristotle, plane geometry is about properties of certain sensible objects—specifically, dimensional continuity—and certain properties possessed by actual and potential compass-and-straightedge drawings (...) qua quantitative and continuous. For their part, the objects of stereometry are potential sensible three-dimensional figures qua quantitative and continuous. (shrink)

The interpretation of Parmenides’ Περί Φύσεως is a fascinating topic to which philosophers, historians of philosophy and scientists have dedicated many studies along the history of Western thought. The aim of this paper is to present the reading of Parmenides’s work offered by Federigo Enriques. It is based on several original theses: (1) Parmenides was the discoverer of abstract geometry; (2) his critics was addressed against the Pythagoreans rather than against Heraclitus; (3) Parmenides discovered and applied the contradiction and (...) the third excluded principles in the context of his research on foundation of geometry; (4) Parmenides’s metaphysical and physical conceptions have their bases in his speculation on geometry; (5) Parmenides used the principle of sufficient reason. Enriques’s reading is worth being expounded and discussed within the historical, philosophical and scientific context in which it is inserted. Since Enriques’s ideas are not widely known and discussed, my research has the purpose to fill this gap. The article also aims to provide elements to illustrate the discussion on Parmenides in the first half of the last century. (shrink)

Open Court began publishingThe Monistin 1890 as a journal“devotedto the philosophy of science”that regularly included mathematics. The audiencewas understood to be“cultured people who have not a technical mathematicaltraining”but nevertheless“have a mathematical penchant.”With these constraints,the mathematical content varied from recreations to logical foundations, but every-one had something to say about non-Euclidean geometry, in debates that rangedfrom psychology to semantics. The focus in this essay is on the contested value ofmathematical expertise in legitimating what should be considered as mathematics.While some mathematicians (...) urgedThe Monistto uphold disciplinary standards ofgeometrical reasoning, authors opposed tonon-Euclidean geometry aligned theirreasoning with practical applications, universal know-how, and nonhierarchicaldemocracy. As one contributorinquired,“How isthe professional expert betterfit-ted to see more lucidly in dealing with the elements of geometry than any otherperson of good geometric faculty?”. (shrink)

Edmund Husserl has argued that we can intuit essences and, moreover, that it is possible to formulate a method for intuiting essences. Husserl calls this method 'ideation'. In this paper I bring a fresh perspective to bear on these claims by illustrating them in connection with some examples from modern pure geometry. I follow Husserl in describing geometric essences as invariants through different types of free variations and I then link this to the mapping out of geometric invariants in (...) modern mathematics. This view leads naturally to different types of spatial ontologies and it can be used to shed light on Husserl's general claim that there are different ontologies in the eidetic sciences that can be systematically related to one another. The paper is rounded out with a consideration of the role of ideation in the origins of modern geometry, and with a brief discussion of the use of ideation outside of pure geometry. (shrink)

What if gravity satisfied the Klein-Gordon equation? Both particle physics from the 1920s-30s and the 1890s Neumann-Seeliger modification of Newtonian gravity with exponential decay suggest considering a "graviton mass term" for gravity, which is _algebraic_ in the potential. Unlike Nordström's "massless" theory, massive scalar gravity is strictly special relativistic in the sense of being invariant under the Poincaré group but not the 15-parameter Bateman-Cunningham conformal group. It therefore exhibits the whole of Minkowski space-time structure, albeit only indirectly concerning volumes. Massive (...) scalar gravity is plausible in terms of relativistic field theory, while violating most interesting versions of Einstein's principles of general covariance, general relativity, equivalence, and Mach. Geometry is a poor guide to understanding massive scalar gravity: matter sees a conformally flat metric due to universal coupling, but gravity also sees the rest of the flat metric in the mass term. What is the 'true' geometry, one might wonder, in line with Poincaré's modal conventionality argument? Infinitely many theories exhibit this bimetric 'geometry,' all with the total stress-energy's trace as source; thus geometry does not explain the field equations. The irrelevance of the Ehlers-Pirani-Schild construction to a critique of conventionalism becomes evident when multi-geometry theories are contemplated. Much as Seeliger envisaged, the smooth massless limit indicates underdetermination of theories by data between massless and massive scalar gravities---indeed an unconceived alternative. At least one version easily could have been developed before General Relativity; it then would have motivated thinking of Einstein's equations along the lines of Einstein's newly re-appreciated "physical strategy" and particle physics and would have suggested a rivalry from massive spin 2 variants of General Relativity. The Putnam-Grünbaum debate on conventionality is revisited with an emphasis on the broad modal scope of conventionalist views. Massive scalar gravity thus contributes to a historically plausible rational reconstruction of much of 20th-21st century space-time philosophy in the light of particle physics. An appendix reconsiders the Malament-Weatherall-Manchak conformal restriction of conventionality and constructs the 'universal force' influencing the causal structure. Subsequent works will discuss how massive gravity could have provided a template for a more Kant-friendly space-time theory that would have blocked Moritz Schlick's supposed refutation of synthetic _a priori_ knowledge, and how Einstein's false analogy between the Neumann-Seeliger-Einstein modification of Newtonian gravity and the cosmological constant \Lambda generated lasting confusion that obscured massive gravity as a conceptual possibility. (shrink)

Twentieth-century developments in logic and mathematics have led many people to view Euclid’s proofs as inherently informal, especially due to the use of diagrams in proofs. In _Euclid and His Twentieth-Century Rivals_, Nathaniel Miller discusses the history of diagrams in Euclidean Geometry, develops a formal system for working with them, and concludes that they can indeed be used rigorously. Miller also introduces a diagrammatic computer proof system, based on this formal system. This volume will be of interest to mathematicians, (...) computer scientists, and anyone interested in the use of diagrams in geometry. (shrink)

Suppose your evil sibling travels back in time, intending to lethally poison your grandfather during his infancy. Determined to save grandpa, you grab two antidotes and follow your sibling through the wormhole. Under normal circumstances, each antidote has a 50% chance of curing a poisoning. Upon finding young grandpa, poisoned, you administer the first antidote. Alas, it has no effect. The second antidote is your last hope. You administer it---and success: the paleness vanishes from grandpa's face, he is healed. As (...) you administered the first antidote, what was the chance that it would be effective? -/- This essay offers a systematic account of this case, and others like it. The central question is this: Given a certain time travel structure, what are the chances? In particular, I'll develop a theory about the connection between these chances and the chances in ordinary, time-travel-free contexts. Central to the account is a Markov condition involving the boundaries of spacetime regions. (shrink)

The standard mathematical account of the sub-metrical geometry of a space employs topology, whose foundational concept is the open set. This proves to be an unhappy choice for discrete spaces, and offers no insight into the physical origin of geometrical structure. I outline an alternative, the Theory of Linear Structures, whose foundational concept is the line. Application to Relativistic space-time reveals that the whole geometry of space-time derives from temporal structure. In this sense, instead of spatializing time, Relativity (...) temporalizes space. (shrink)

This paper gives a contextualized reading of Kant's theory of real definitions in geometry. Though Leibniz, Wolff, Lambert and Kant all believe that definitions in geometry must be ‘real’, they disagree about what a real definition is. These disagreements are made vivid by looking at two of Euclid's definitions. I argue that Kant accepted Euclid's definition of circle and rejected his definition of parallel lines because his conception of mathematics placed uniquely stringent requirements on real definitions in (...) class='Hi'>geometry. Leibniz, Wolff and Lambert thus accept definitions that Kant rejects because they assign weaker roles to real definitions. (shrink)

Gothic cathedrals like Chartres were built in a discontinuous process by groups of masons using their own local knowledge, measures, and techniques. They had neither plans nor knowledge of structural mechanics. The success of the masons in building such large complex innovative structures lies in the use of templates, string, constructive geometry, and social organization to assemble a coherent whole from the messy heterogeneous practices of diverse groups of workers. Chartres resulted from the ad hoc accumulation of the work (...) of many men. (shrink)

Disoriented animals from ants to humans reorient in accord with the shape of the surrounding surface layout: a behavioral pattern long taken as evidence for sensitivity to layout geometry. Recent computational models suggest, however, that the reorientation process may not depend on geometrical analyses but instead on the matching of brightness contours in 2D images of the environment. Here we test this suggestion by investigating young children's reorientation in enclosed environments. Children reoriented by extremely subtle geometric properties of the (...) 3D layout: bumps and ridges that protruded only slightly off the floor, producing edges with low contrast. Moreover, children failed to reorient by prominent brightness contours in continuous layouts with no distinctive 3D structure. The findings provide evidence that geometric layout representations support children's reorientation. (shrink)

The electronic and muonic hydrogen energy levels are calculated very accurately [1] in Quantum Electrodynamics (QED) by coupling the Dirac Equation four vector (c ,mc2) current covariantly with the external electromagnetic (EM) field four vector in QED’s Interactive Representation (IR). The c -Non Exclusion Principle(c -NEP) states that, if one accepts c as the electron/muon velocity operator because of the very accurate hydrogen energy levels calculated, the one must also accept the resulting electron/muon internal spatial and time coordinate operators (ISaTCO) (...) derived directly from c without any assumptions. This paper does not change any of the accurate QED calculations of hydrogen’s energy levels, given the simplistic model of the proton used in these calculations [1]. The Proton Radius Puzzle [2, 3] may indicate that new physics is necessary beyond the Standard Model (SM), and this paper describes the bizarre, and very different, situation when the electron and muon are located “inside” the spatially extended proton with their Centers of Charge (CoCs) orbiting the proton at the speed of light in S energy states. The electron/muon center of charge (CoC) is a structureless point that vibrates rapidly in its inseparable, non-random vacuum whose geometry and time structure are defined by the electron/muon ISaTCO discrete geometry. The electron/muon self mass becomes finite in a natural way due to the ISaTCOs cutting off high virtual photon energies in the photon propagator. The Dirac-Maxwell-Wilson (DMW) Equations are derived from the ISaTCO for the EM fields of an electron/muon, and the electron/muon “look” like point particles in far field scattering experiments in the same way the electric field from a sphere with evenly distributed charge “e” “looks” like a point with the same charge in the far field (Gauss Law). The electron’s/muon’s three fluctuating CoC internal spatial coordinate operators have eight possible eigenvalues [4, 5, 6] that fall on a spherical shell centered on the electron’s CoM with radius in the rest frame. The electron/muon internal time operator is discrete, describes the rapid virtual electron/positron pair production and annihilation. The ISaTCO together create a current that produce spin and magnetic moment operators, and the electron and muon no longer have “intrinsic” properties since the ISaTCO kinematics define spin and magnetic moment properties. (shrink)

It has become a fact that fluency and competency in utilizing the advancement of technology, specifically the computer and the internet is one way that could help in facilitating learning in mathematics. This study investigated the effects of Dynamic Geometry Software (DGS) and Computer Algebra Systems (CAS) in teaching Mathematics. This was conducted in Zamboanga del Sur National High School (ZSNHS) during the third grading period of the school year 2015-2016. The study compared the achievement and attitude towards Mathematics (...) between those students taught with DGS and CAS and those students who were taught without DGS and CAS. A quasi-experimental pretest-posttest control group design was used with two groups which was matched according to their mathematical level. An achievement test was developed and administered to assess the students’ achievement in Mathematics. Aiken’s attitude scale was also used to assess the students’ attitude towards Mathematics. The data were treated with One-Way Analysis of Covariance (ANCOVA) and t-test at 0.05 level of significance. The study revealed a significant difference between the mathematics achievement between students taught with DGS and CAS and those who were taught without it. The integration of DGS and CAS in Mathematics lessons leads to better academic achievement result compared with that of the conventional method of teaching. Moreover, there was no significant difference on the attitude level of the students between the two groups. (shrink)