This volume contains four chapters, of which the first treats of steps; the second of rotations, turns, arc-steps; the third of quaternions; and the fourth of equations of the first degree.
It presents a good elementary exposition of the method of quaternions, as taught by Hamilton and Tait. The student who masters it will find that he has acquired some real knowledge, not merely additional dexterity in formal manipulations. At the same time it must be said that he will encounter almost all of the difficulties, anomalies, and imperfections which have long obscured the method, and prevented its general acceptance and development.
In past years it had been maintained, in opposition to Hamilton and Tait, that a vector is a vector, not a right quaternion. Arthur s. Hathaway recognizes the distinction, and proposed to restrict vector to mean right quaternion, while using 'step' to denote a directed length. This is a proposal which cannot be commended; for writers agree in using the word vector to mean a quantity which has linear direction. The confusion is introduced by the false identification on the part of the quaternionists of the vector with the right quaternion. The former has linear direction, the latter angular direction. The plain principle of procedure is to leave 'vector' to its established use; and, if necessary, to devise some other word for "right quaternion."
William Rowan Hamilton laid great stress on his analysis of a quaternion into the sum of a number and a line. From the preface (p. vi) we observe that the author views the whole quaternion as a number. At p. 53 we have the proposition, "The number 'r' rotated through '2 arc q' is the number qrqˉ¹." With Hamilton "number" meant a quantity destitute of direction; with the mathematical physicist it means a quantity destitute of physical dimensions; but what meaning are we to attach to it in order that it may be capable of rotation through an angle?
A quaternion is not a number in Hamilton's sense of the word, nor is it always a number in the sense understood by the mathematical physicist. The whole resistance in an alternating circuit is the geometrical sum of an ohmic resistance and an inductive resistance; it is a quaternion, but the dimensions are not zero. To explain the quaternion by means of the complex number is to take a step backwards; for Hamilton strove in vain to extend algebra to space until he gave up the time or number idea, and gave the whole a geometrical basis.
In physical science, both the scalar product and the vector product of two vectors are of frequent occurrence; for instance, work is the scalar product of a force and the displacement of its point of application; electromotive force per unit of length is the vector product of the velocity of the conductor and the density of the magnetic flux.
The Hamiltonian theory attempts to explain 'AB', the vector product of two vectors, by regarding 'A' as an operator on 'B'. But the explanation given does not explain the scalar product; and, from the physical point of view, it is evident that the two vectors enter symmetrically into the product. The operator and operand theory is entirely inadequate to explain the products which occur in mathematical physics. Arthur Hathaway gets over the difficulty by omitting the multiplication of vectors ("steps" in his nomenclature), and treating only of the products of right quaternions.
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It presents a good elementary exposition of the method of quaternions, as taught by Hamilton and Tait. The student who masters it will find that he has acquired some real knowledge, not merely additional dexterity in formal manipulations. At the same time it must be said that he will encounter almost all of the difficulties, anomalies, and imperfections which have long obscured the method, and prevented its general acceptance and development.
In past years it had been maintained, in opposition to Hamilton and Tait, that a vector is a vector, not a right quaternion. Arthur s. Hathaway recognizes the distinction, and proposed to restrict vector to mean right quaternion, while using 'step' to denote a directed length. This is a proposal which cannot be commended; for writers agree in using the word vector to mean a quantity which has linear direction. The confusion is introduced by the false identification on the part of the quaternionists of the vector with the right quaternion. The former has linear direction, the latter angular direction. The plain principle of procedure is to leave 'vector' to its established use; and, if necessary, to devise some other word for "right quaternion."
William Rowan Hamilton laid great stress on his analysis of a quaternion into the sum of a number and a line. From the preface (p. vi) we observe that the author views the whole quaternion as a number. At p. 53 we have the proposition, "The number 'r' rotated through '2 arc q' is the number qrqˉ¹." With Hamilton "number" meant a quantity destitute of direction; with the mathematical physicist it means a quantity destitute of physical dimensions; but what meaning are we to attach to it in order that it may be capable of rotation through an angle?
A quaternion is not a number in Hamilton's sense of the word, nor is it always a number in the sense understood by the mathematical physicist. The whole resistance in an alternating circuit is the geometrical sum of an ohmic resistance and an inductive resistance; it is a quaternion, but the dimensions are not zero. To explain the quaternion by means of the complex number is to take a step backwards; for Hamilton strove in vain to extend algebra to space until he gave up the time or number idea, and gave the whole a geometrical basis.
In physical science, both the scalar product and the vector product of two vectors are of frequent occurrence; for instance, work is the scalar product of a force and the displacement of its point of application; electromotive force per unit of length is the vector product of the velocity of the conductor and the density of the magnetic flux.
The Hamiltonian theory attempts to explain 'AB', the vector product of two vectors, by regarding 'A' as an operator on 'B'. But the explanation given does not explain the scalar product; and, from the physical point of view, it is evident that the two vectors enter symmetrically into the product. The operator and operand theory is entirely inadequate to explain the products which occur in mathematical physics. Arthur Hathaway gets over the difficulty by omitting the multiplication of vectors ("steps" in his nomenclature), and treating only of the products of right quaternions.
A PRIMER OF QUATERNIONS
This volume contains four chapters, of which the first treats of steps; the second of rotations, turns, arc-steps; the third of quaternions; and the fourth of equations of the first degree.
It presents a good elementary exposition of the method of quaternions, as taught by Hamilton and Tait. The student who masters it will find that he has acquired some real knowledge, not merely additional dexterity in formal manipulations. At the same time it must be said that he will encounter almost all of the difficulties, anomalies, and imperfections which have long obscured the method, and prevented its general acceptance and development.
In past years it had been maintained, in opposition to Hamilton and Tait, that a vector is a vector, not a right quaternion. Arthur s. Hathaway recognizes the distinction, and proposed to restrict vector to mean right quaternion, while using 'step' to denote a directed length. This is a proposal which cannot be commended; for writers agree in using the word vector to mean a quantity which has linear direction. The confusion is introduced by the false identification on the part of the quaternionists of the vector with the right quaternion. The former has linear direction, the latter angular direction. The plain principle of procedure is to leave 'vector' to its established use; and, if necessary, to devise some other word for "right quaternion."
William Rowan Hamilton laid great stress on his analysis of a quaternion into the sum of a number and a line. From the preface (p. vi) we observe that the author views the whole quaternion as a number. At p. 53 we have the proposition, "The number 'r' rotated through '2 arc q' is the number qrqˉ¹." With Hamilton "number" meant a quantity destitute of direction; with the mathematical physicist it means a quantity destitute of physical dimensions; but what meaning are we to attach to it in order that it may be capable of rotation through an angle?
A quaternion is not a number in Hamilton's sense of the word, nor is it always a number in the sense understood by the mathematical physicist. The whole resistance in an alternating circuit is the geometrical sum of an ohmic resistance and an inductive resistance; it is a quaternion, but the dimensions are not zero. To explain the quaternion by means of the complex number is to take a step backwards; for Hamilton strove in vain to extend algebra to space until he gave up the time or number idea, and gave the whole a geometrical basis.
In physical science, both the scalar product and the vector product of two vectors are of frequent occurrence; for instance, work is the scalar product of a force and the displacement of its point of application; electromotive force per unit of length is the vector product of the velocity of the conductor and the density of the magnetic flux.
The Hamiltonian theory attempts to explain 'AB', the vector product of two vectors, by regarding 'A' as an operator on 'B'. But the explanation given does not explain the scalar product; and, from the physical point of view, it is evident that the two vectors enter symmetrically into the product. The operator and operand theory is entirely inadequate to explain the products which occur in mathematical physics. Arthur Hathaway gets over the difficulty by omitting the multiplication of vectors ("steps" in his nomenclature), and treating only of the products of right quaternions.
It presents a good elementary exposition of the method of quaternions, as taught by Hamilton and Tait. The student who masters it will find that he has acquired some real knowledge, not merely additional dexterity in formal manipulations. At the same time it must be said that he will encounter almost all of the difficulties, anomalies, and imperfections which have long obscured the method, and prevented its general acceptance and development.
In past years it had been maintained, in opposition to Hamilton and Tait, that a vector is a vector, not a right quaternion. Arthur s. Hathaway recognizes the distinction, and proposed to restrict vector to mean right quaternion, while using 'step' to denote a directed length. This is a proposal which cannot be commended; for writers agree in using the word vector to mean a quantity which has linear direction. The confusion is introduced by the false identification on the part of the quaternionists of the vector with the right quaternion. The former has linear direction, the latter angular direction. The plain principle of procedure is to leave 'vector' to its established use; and, if necessary, to devise some other word for "right quaternion."
William Rowan Hamilton laid great stress on his analysis of a quaternion into the sum of a number and a line. From the preface (p. vi) we observe that the author views the whole quaternion as a number. At p. 53 we have the proposition, "The number 'r' rotated through '2 arc q' is the number qrqˉ¹." With Hamilton "number" meant a quantity destitute of direction; with the mathematical physicist it means a quantity destitute of physical dimensions; but what meaning are we to attach to it in order that it may be capable of rotation through an angle?
A quaternion is not a number in Hamilton's sense of the word, nor is it always a number in the sense understood by the mathematical physicist. The whole resistance in an alternating circuit is the geometrical sum of an ohmic resistance and an inductive resistance; it is a quaternion, but the dimensions are not zero. To explain the quaternion by means of the complex number is to take a step backwards; for Hamilton strove in vain to extend algebra to space until he gave up the time or number idea, and gave the whole a geometrical basis.
In physical science, both the scalar product and the vector product of two vectors are of frequent occurrence; for instance, work is the scalar product of a force and the displacement of its point of application; electromotive force per unit of length is the vector product of the velocity of the conductor and the density of the magnetic flux.
The Hamiltonian theory attempts to explain 'AB', the vector product of two vectors, by regarding 'A' as an operator on 'B'. But the explanation given does not explain the scalar product; and, from the physical point of view, it is evident that the two vectors enter symmetrically into the product. The operator and operand theory is entirely inadequate to explain the products which occur in mathematical physics. Arthur Hathaway gets over the difficulty by omitting the multiplication of vectors ("steps" in his nomenclature), and treating only of the products of right quaternions.
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A PRIMER OF QUATERNIONS
A PRIMER OF QUATERNIONS
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Product Details
| BN ID: | 2940016322025 |
|---|---|
| Publisher: | OGB |
| Publication date: | 03/05/2013 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| File size: | 8 MB |
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