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The Positive Sciences of the Ancient Hindus c5: Chapter II. Hindu Ideas on Mechanics (Kinetics)

The Positive Sciences of the Ancient Hindus c5

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Chapter II. Hindu Ideas on Mechanics (Kinetics)

Section 1. Analysis of Motion

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In his Bhâṣya¹ on the Vaiśeṣika¹ Aphorisms, written probably in the third or fourth century of the Christian era, if not earlier, Praśastapâda begins the Section on Motion (कर्मग्रन्थ) with the definition of *karma* (motion, *lit.* work) as the unconditional cause of conjunction and disjunction, *i.e.* of change of place in a particle (संयोगविभागनिरपेक्षकारणं). He regards *karma* (motion) as instantaneous (क्षणिक) in its simplest form, distinguishing it from *vega* (impressed motion, momentum), which is a persistent tendency, Sanskâra, and implies a series of motions. Accordingly in one and the same particle there can be only one motion (*karma*) at any given moment, since its change of place at that moment is one and definite (एकदा एकस्मिन् द्रव्ये एकमेव कर्म्म वर्तते। प्रशस्तपादभाष्य, कर्म्म पदार्थनिरूपणं). The supposition of two (instantaneous) motions in the same particle is superfluous. They may be so opposed as to neutralise each other, in which case the particle would be at rest. If they are not so opposed, and motion (*i.e.* an instantaneous change of place) follows, then, since this change of place is a definite change, one motion would be sufficient to account for it, and the hypothesis of two motions would be meaningless (अथ अविरुद्धकर्म्मद्दयसमावेशः तदा एकस्मादेव उपपत्तेः द्वितीयकल्पनावैय्यर्थम्—Praśastapâda). One and the same motion can effect only one particle, as the changes of place of different particles must be different (एकं कर्म्म न अनेकत्र वर्त्तते—*Ibid.*).

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¹ The system of transliteration adopted in Chapters II.–VI. differs from that adopted in Chapter I. and Chapter VII.

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Now motion is always marked by a certain direction (दिग्विजयिकयोरम्भकत्वमस्य विशेषः).

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(i) The successive motions of a particle may be in the same direction (rectilinear), *e.g.* (*a*) upward or downward vertical motion, as in throwing upwards or downwards in the case of objects moved by volition directly or indirectly (उत्क्षेपणं, अपक्षेपणं), or (*b*) other forms of rectilinear motion, contraction, dilation (आकुञ्चनं, प्रसारणं); or

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(ii) the directions of the successive motions may be different, as in curvilinear motion (यदनियतदिक्प्रदेशसंयोगविभागकारणं तद्गमनं), *e.g.* भ्रमण (rotatory motion), स्पन्दन (vibratory motion), etc. All these are varieties of Gamana (गमन, curvilinear motion). उत्क्षेपणादिशब्दैः अनवरुद्धानां भ्रमणपतनस्पन्दनादीनां अवरोधार्थं गमनग्रहणं कृतम्—*Ibid.*; *cf.* Śankara Miśra, गमनत्वं च जातिविशेषः भ्रमणरेचनस्यन्दनोर्द्ध्वज्वलननमनोन्नमनादिष्वपि. In another sense, all kinds of motion in material (inanimate) objects, whether rectilinear or curvilinear, are called Gamana (गमन).

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(एतत् पञ्चविधमपि कर्म्म शरीरावयवेषु च तत् सम्बद्धेषु तत् प्रत्ययं चासत्प्रत्ययं च । यत् अन्यत् अप्रत्ययमेव तेषु अन्येषु च तत् गमनमिति Praśastapâda, कर्म्मग्रन्थ) ।

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Single particles, then, may have a serial motion. When particles (अवयवाः) combine to form a body (अवयवी), they may move continuously in a straight line, in which case the body is said to move in that direction (अवयविक्रियाया यावदवयवक्रियानिबन्धनत्वात्, the action of a composite whole is determined by the action of the constituent parts taken together). But different particles may move in different directions, or again, the particles may have a curvilinear motion, and in such cases it appears as if different motions are impressed on the body, *e.g.* the falling leaf driven by the wind may have a rotatory or vibratory motion (भ्रमण, स्पन्दन) and a vertical downward motion (पतन) at the same time. Here each particle of the leaf taken separately has only one motion or change of place at the same moment, but from the point of view of the observer द्रष्टा, the particles have a rotatory or vibratory motion in one relation, and the leaf as a whole has a downward motion in another relation. The motion at any instant is really one, but for convenience of analysis we consider the rotatory change of place as separate from the change of place in the downward direction. एकस्मिन् कर्मणि युगपत् द्रष्टृणां भ्रमणपतनप्रवेषनप्रत्यया: कथं भवन्ति । अत्र ब्रूम अवयवावयविनो दिग्विश्लिष्टसंयोगविभागानां भेदात्—*Ibid.*

Section 2. Motion Considered in Relation to its Causes

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Various kinds of motion are observed:—

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I. Movements which are not caused by contact with matter:—

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(1) Movements caused by volition (प्रयत्न), *e.g.* the movement of the hand.

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(2) Movement as of a falling body. This is caused by gravity (गुरुत्व), which in the astronomical treatises of Âryabhata, Brahmagupta, and Bhâskara is ascribed to the attraction (आकर्षण, pulling force) exercised by the earth on a material body. The force of gravity may be counteracted by volition (विधारकप्रयत्न), as in holding up the hand, or by contact, as when a body rests on a support, or by Vega (वेग), impressed motion, as in the flying arrow which is kept from falling by the motion impressed on it.

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(3) Motion of fluids, as the downward flow in a stream (स्यन्दन). This is due to fluidity (द्रवत्व), which is a characteristic property of certain kinds of atoms, in some cases original, in others produced by the contact of these atoms with the atomic particles of heat (*e.g.* in the fire). But Śaṅkara Miśra points out that this property, fluidity, is only a concomitant condition (असमवायिकारण); the efficient cause (निमित्तकारण) is even in this case gravity (गुरुत्व) in the particles of the fluid. यत्प्रदूरसंसरखं स्यन्दनं तत् द्रवत्वात् असमवायिकारणात् उत्पद्यते, गुरुत्वात् निमित्तकारणात् असमवायिकारणेषु (Śaṅkara Miśra, Upaskâra, on Sûtra 4, Âhnika 2, Chap. V., of the Vaiśeṣika Sûtras).

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(4) Certain motions, not due to material contact, of which the mechanical causes are unknown, and which may be ascribed to the universal final cause, Adṛiṣṭa (अदृष्ट), *e.g.* the first motion in atoms at the beginning of Creation, the upward motion of fiery particles or atoms, and the oblique motion of gaseous particles, *vâyu* (वायु), the movement of iron towards the magnet, capillary motion (अभिसर्पण) as of liquid particles from the root to the stem of a plant. The upward motions (चारोहण) of water-particles in evaporation and in boiling do not require the hypothesis of Adṛiṣṭa, as these are caused by the pressure of heat-corpuscles (तेजः परमाणु) and the contact with air-particles (नोदनापीड़नात् संयुक्तसंयोगाच्च—Vaiśeṣika Sûtra, Chap. V., Âhnika 2).

Meaning of Adriṣṭa (अदृष्ट)

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*Adṛiṣṭa* (*lit.* unseen) stands for "unknown cause" or "unexplained Nature" in the earlier Vaiśeṣika writers. Several classes of cases falling under Adṛiṣṭa are distinguished, *e.g.*—

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(1) The operation of merit and demerit (धर्म and अधर्म), a transcendental cause, which has to be posited in explaining the conjunctions and disjunctions of souls with their organic vehicles (bodies), which cannot be ascribed to natural causes, but presuppose the law of Karma, or the operation of moral causation, as superimposed on the natural order.

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(2) Various kinds of motion in the different classes of elements, *e.g.* their natural modes of operation, such as the dispersion of *vâyu* (air, gas), the upward motion of fire, the attraction of the needle by the magnet, etc., motions which serve the ends of Creation and of created beings (उपकारकं अपकारकं). Such natural properties must be ascribed to Adṛiṣṭa, final causality (उपकारापकारसमर्थं अदृष्टकारितं), provided the cause cannot be ascertained by *observation or inference.*

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¹ इष्टप्रवृत्तिभ्याम् धर्माधर्माभ्याम् कर्मात्पद्यते—*Ibid.* अपसर्पणकर्म उपसर्पणकर्म, etc.

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² एवमन्यदपि महाभूतेषु यत्प्रत्यक्षानुमानाभ्यामनुपलभ्यमानकारणं उपकारापकारसमर्थं च भवति तदपि अदृष्टकारितं—*Ibid.*

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Jayanta in the Nyâya-Manjarî notes that Adṛiṣṭa is resorted to in explanation of observed phenomena only when these cannot be derived in any way from the operation of known causes—यदि अदृष्टमन्तरेण दृष्टं न सिद्ध्यति काममदृष्टं कल्यताम् अन्यथा अपितु तदुपपत्तौ किं तदुपकल्पनेन । दृष्टसिद्धयेहि अदृष्टं कल्प्यते नतु दृष्टविघाताय । न्यायमञ्जरी—Âhnika 1.

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Similarly, Jayanta notes that, when anything is put down as natural (स्वाभाविक), we may mean either that it has no cause or no uniform cause, or no known cause. Of these the first two alternatives must always be dismissed ; and “natural” can only mean something of which the cause has not yet been ascertained (स्वाभाविकं नाम किमुच्यते किमहेतुकं अविज्ञातहेतुकं अनियतहेतुकं वा । न तावदहेतुकं कार्यं सम्भवति ........ नापि अनियतहेतुकं कार्यं किञ्चिदस्ति – न्यायमञ्जरी – चार्वाकमतनिरास ।)

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This sound interpretation of Adṛiṣṭa was afterwards obscured. In modern writers of the Nyâya-Vaiśeṣika school, physical and mechanical ideas have suffered a set-back, and even the formation of the hail-stone is ascribed to the operation of moral causes (धर्माधर्म).

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II. We come now to motions produced by contact (संयोग). Such motions may be classed as follows, according to the nature of the contact originating them :—

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(1) Motion due to direct contact with a body exercising continued pressure (नोदन), *e.g.* the motion of an object pushed or pulled by the hand, the motion of the mud under heavy stones, the motion of the arrow due to the pressure exercised by the bow-string, the motion of the bow-string due to the pressure of the elastic bow as it recovers its original shape, the motion of clouds, of volumes of dust, of balloons, sailing vessels and other vehicles under the impelling force (pressure, नोदन, प्रेरण) of the wind, etc. वायुर्मेघादिप्रेरणधारणादिसमर्थः Praśastapâda, वायुनिरूपण-मेघादोत्यादिपदेन यानपोतादिपरिग्रहस्तेषामपि वायुना प्रेर्य्यमाणत्वात् । Śrîdhara मेघादीति चादिग्रहणात् वेहायसानां विमानादीनां (balloons in the sky) भौमादीनां च पानपात्रपांशुपटलादीनां जलानलयोश्च परिग्रहः (Udayana, Kiraṇâvalî, वायुनिरूपणम् ).

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*N. B.*—Udayana makes a similar reference to balloons filled with gas or smoke (धूमापूरितचर्म्मपुटकं) in discussing the opinion that air has weight (Kiraṇâvalî, वायुनिरूपणम्). These passages show that balloons were known in Udayana’s time (970 a.d.—*vide* उदयन, Lakṣaṇâvalî).

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(2) Motion due to direct contact for an instant with a body that strikes and produces an impact (अभिघात), *e.g.* in the cases of a stone falling against a hard object (पाषाणादिषु निष्ठुरेवस्तुनि अभिपतितेषु—Śrîdhara), the potter’s rod striking the wheel, the mortar struck against the pestle. Instantaneous disjunction is necessary to impact. If there is continued contact, the result is pressure (नोदन). In some cases there may be disjunction (*i.e.* a rebound) after continued pressure (यत्र अभिघातकं द्रव्यं भूमदेशं अभिहत्य किञ्चिदधो नोत्वा उत्पतति—श्रीधर—न्यायकन्दली).

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(3) Motion due to direct contact with an elastic body which exercises a moving force by means of its elasticity (स्थितिस्थापकत्व) in the act of restitution of the original form (यथास्थितं स्थापयति), *e.g.* the motion of the bow-string due to the force exercised by the piece of bamboo (the bent bow). This force of restitution in an elastic body is a kind of *sanskâra*, *i.e.* persistent tendency (धनुःशाखाशृङ्गदन्तास्थिसूत्रवस्त्रादिषु भुग्नसंवर्त्तितेषु स्थितिस्थापकत्वकार्य्यं संलक्ष्यते—प्रशस्तपादभाष्य, bows, twigs, tooth-bones, horn, thread, cloth, etc., are noted as elastic).

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(4) Motion due to contact with a body which is itself in contact with another which possesses Vega (impressed motion or momentum) (वेगवद्द्रव्यसंयुक्तसंयोग). This is the fact of the transmission of pressure or moving force, and the consequent production or communication of motion, as, for example, in the pulling of an object by means of a string, the pushing of the potter's wheel by the potter's rod, etc.

Section 3. Cause of Motion, or Force

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Force is of the following kinds:—
1. Continued pressure (नोदन).
2. Impact (अभिघात).
3. Persistent tendency (संस्कार), of which there are two kinds: (*a*) Vega (वेग, impressed motion, momentum), the persistent tendency to motion in a moving body, and (*b*) the tendency to restitution of shape in an elastic body (स्थितिस्थापकसंस्कार).

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*N.B.*—The psychical *sanskâra* (भावना) is here omitted.

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4. Transmitted force, as in pulling by a string, pushing by a rod, etc.
5. Gravity.
6. Fluidity.
7. Volition.
8. Adṛiṣṭa, comprising various unknown agencies.

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In every case of motion produced by contact, Vega is a contributory cause, as the body originating the motion must possess Vega (impressed motion, momentum).

The Concept of Vega (वेगाख्यसंस्कार)

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A motion (*karma*) being conceived as a change of place in a particle is held to be incapable of producing another motion; but the pressure, impact, or other force which produces the first motion produces through that motion a *sanskâra* or persistent tendency to motion (Vega), which is the cause of continued motion in a straight line, *i.e.* in the direction of the first motion (नियतदिक्रिया प्रबन्धहेतुः यद्दिगाभिमुख्येन क्रियावेगो जन्यते तद्दिशभिमुख्येनैव क्रियासन्तानस्य हेतुरिति—श्रीधर, न्याय-कन्दली, संस्कार-निरूपणम्).

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The Vaiśeṣikas accept one *sanskâra* (impressed motion, momentum) lasting till the cessation of the motion. Udyotakara and other writers of the Nyâya school suppose a series of *sanskâras*, each generating the one that succeeds it. (संस्कारेण उत्तरोत्तरकर्म्मसन्तानो जायते अनन्योत्तरसंयोगेन कर्म्मणि नष्टे संस्कारेण कर्म्मान्तरजननात् एक एव संस्कारः कर्म्मसन्तानजनकः नतु कर्म्मसन्तानवत् संस्कारसन्तानोऽपि अभ्युपगन्तुमुचितः—शङ्करमिश्र।) It will be seen that the Nyâya view is adequate to explain acceleration, which it logically implies. The force of *sanskâra* (शक्ति) diminishes by doing work (कार्य्यकरणात्) against a counteracting force, and when the *sanskâra* is in this way entirely destroyed, the moving body comes to rest (संस्कारो यावत्पतनमनुवर्त्तते। यथा यथा चास्य कार्य्यकरणात् शक्तिः क्षीयते तथा तथा कार्य्यमन्दतरतमादि-भेदभिन्नमुपजायते—श्रीधर—न्याय-कन्दली-कर्म्मग्रन्थ). Vega, it will be seen, corresponds to inertia in some respects, and to momentum (impressed motion) in others. This is the nearest approach to Newton's First Law of Motion in the Vaiśeṣika theory of motion.

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Vega (impressed motion), or this tendency to move on in a straight line, is counteracted by contact with tangible objects (स्पर्शवद्द्रव्यसंयोग), *e.g.* by impact or friction, including friction with the still atmosphere (स्तिमितवायु), as in the case of the arrow.

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Vega (momentum) produces work in opposition to the resisting force, and thereby becomes weaker and weaker (मन्दतर मन्दतम) until it comes to an end (तत्र वेगो मूर्त्तिमत्सु पञ्चसु द्रव्येषु निमित्तविशेषापेक्षात् कर्म्मणो जायते, नियत-दिक्-क्रिया-प्रबन्धहेतुः स्पर्शवद्द्रव्यसंयोग-विशेषविरोधी Praśastapâda, संस्कार-निरूपणम्—मन्दस्तु वेगः स्पर्शवद्द्रव्यसंयोगमात्रेण विनश्यति यथा अतिदूरगतस्य इषोः स्तिमितवायु-प्रतिबद्धस्य—श्रीधर, संस्कारनिरूपणम्).

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*Causes of Pressure* (नोदन) *and of Impact* (अभिघात).

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Pressure is produced by contact acting in conjunction with Vega (impressed motion), elasticity, gravity, fluidity or volition, *e.g.* the Vega of the wind produces pressure (नोदन) on the grass, that of a current of water on the reed, that of the bow-string on the arrow. Gravity with contact produces pressure, as when the earth sinks under a heavy load.

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Impact is produced by contact with a body possessing Vega (impressed motion) where the contact is instantly followed by disjunction (or rebounding). If the contact continues, the result is pressure (नोदन). (तत्र नोदन गुरुत्व-द्रवत्व-वेग-प्रयत्नान् समस्त-व्यस्तान् अपेक्षामानः यः संयोग-विशेषः नोदनं अविभाग-हेतुः एकस्य कर्म्मणः कारणं तस्मात् चतुर्षु अपि महाभूतेषु कर्म्म भवति . . . . . . वेगापेक्षो यः संयोग-विशेषः विभागहेतुः एकस्य कर्म्मणः कारणं स अभिघात तस्मादपि चतुर्षु-महाभूतेषु कर्म्म भवति—प्रशस्तपाद-भाष्य।)

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It is expressly noted that the four elements, earth, water, air, and fire, are all subject to the forces of pressure and impact. Pressure and impact may be of different degrees (तीव्रमन्दादिभेद). So also Vega (impressed motion, momentum).

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*Illustrations of Combination of Forces.*

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(1) Pressure acting concurrently with impressed motion or Vega, as when the moving hand possessing Vega throws the quoit or a projectile. (पाणिमुक्तेषु गमनविधिः) (ततः संस्कारनोदनाभ्यां तावत् कर्माणि भवन्ति यावत् हस्ततोमर-विभाग इति—प्रशस्तपादभाष्य।)

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Similarly, in the case of the bow-string impelling the arrow, or the potter’s wheel impelled by the rod, the first motion is due to pressure (नोदन), and results in a *sanskâra* (persistent tendency to motion, impressed motion or momentum), but the subsequent motions are produced by the pressure acting concurrently with the *sanskâra* (impressed motion). (तस्मात् संस्कारात् नोदन-सहायात् तावत् कर्माणि भवन्ति यावत् इषुज्या-विभागः—प्रशस्त-पाद, कर्माग्रन्थ—प्रथमं चक्रावयविनि दण्डसंयोगात् कर्म्म उत्पद्यते उत्तरोत्तराणि कर्माणि अभिघातात् कर्मजात् संस्कारात् च भवन्ति दण्डविगमे तु चक्रं तदवयवेषु च संस्कारादेव केवलात्। श्रीधर, न्याय-कन्दलो, कर्माग्रन्थ।)

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(2) Impact (अभिघात) with impressed motion (संस्कार), as when the mortar thrown by the hand rebounds after striking the pestle (संस्कारापेक्षात् अभिघातात् मुसले उत्पतन-कर्म—प्रशस्तपाद भाष्य).

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(3) Pressure (नोदन) acting concurrently with impact (अभिघात), as when the mud sinks when we strike against the ground with the feet. Here, if the feet be not in contact with the mud, but only with the surrounding ground, there is transmitted force (संयुक्तसंयोग).

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(पदादिभिर्नुद्यमानायामभिहन्यमानायां वा पंकाख्यायां पृथिव्यां यत् संयो-गोन्नोदनाभिघातयोरन्यतरापेक्षो उभयापेक्षो वा स संयुक्तसंयोगः। तस्मादपि पृथिव्यादिषु कर्म्म भवति।—Praśastapâda.)

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(4) Gravity concurrently with *sanskâra* or persistent tendency, as in the case of a falling body in the second and following instants. Also the case of a stone thrown against the mud, where gravity (the weight of the stone) combines with the Vega of the stone to produce motion in the mud (Praśastapâda). (आद्यं कर्म्म गुरुत्वात् भवति। तेन कर्म्मणा संस्कारः क्रियते तदनन्तरं उत्तरकर्म्माणि गुरुत्वसंस्काराभ्यां जायते द्वयोरपि प्रत्येकं अन्यत्र सामर्थ्यावधारणात्—श्रीधर, न्याय-कन्दली, कर्म्मग्रन्थे साप्रत्यय-कर्म्मणिरूपणम्।) This case will be further noticed below.

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Udyotakara, the commentator on the Nyâya-Bhâṣya, states that a heavier body falls to the ground with greater Vega (and velocity) than one that is lighter. Udyotakara also holds, and Śrîdhara agrees with him, that the gravity (गुरुत्व) of a body (अवयवी) as a whole composed of particles (अवयवाः) is not the same as the sum of the gravities of the particles. There is a difference in amount which is, however, so small as to be imperceptible. The concept of mass in the New Mechanics of Lorenz may lend some countenance to this curious metaphysical speculation. (Śrîdhara, गुणग्रन्थ, गुरुत्वनिरूपणम्।)

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(5) Volition acting concurrently with gravity, as in lifting up the hand. This is accompanied by transmission when an object, *e.g.* the quoit, is lifted by the hand.

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Sanskâra (impressed motion, momentum), with or without pressure (नोदन) or impact (अभिघात), may be transmitted (दृढ़संयुक्तस्य अवयवस्य उत्तरोत्तरकर्म्माणि संस्कारात् नोदनाच्च अपरेषां संस्कारात् संयुक्तसंयोगाच्च।—Śrîdhara).

Composition of Gravity with Vega (Momentum)

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When a body is let go and falls to the ground, the force acting on it is gravity (गुरुत्व), which the astronomers ascribe to the attraction of the earth. Motion is produced in the first instance by gravity alone, and this leads to a *sanskâra* (impressed motion) in the same direction. But the force of gravity continues to operate, so that, in the moments following the first, the motion is due to gravity as well as *sanskâra*. The resultant motion is one, but both the causes must be conceived as contributing to the resultant. The reason for supposing this combined action is that both gravity and *sanskâra* (impressed motion or momentum) are seen elsewhere to produce motion separately.

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In the case of the falling body, therefore, there is the composition of the two, gravity and Vega, acting in the same direction (उभयसमावेशः) from the second instant onwards. It is as if two motions coalesced and resulted in one.

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Here a good foundation is laid for the explanation of the accelerated motion of falling bodies, but Galileo's discovery was not anticipated, as Galileo's observations and measurements of motion were wanting. (आद्यं कर्म गुरुत्वात् भवति तेन कर्म्मणा संस्कारः क्रियते तदनन्तरं उत्तरकर्म्माणि गुरुत्व-संस्काराभ्याम् जायन्ते द्वयोरपि प्रत्येकं अन्यत्र सामर्थ्यावधारणात्—श्रीधर, कर्म्मग्रंथ—तत् आद्यं गुरुत्वात् द्वितीयादीनि तु गुरुत्वसंस्काराभ्याम्. —प्रशस्तपाद, कर्म्मग्रन्थ ।)

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But in the case of the flying arrow or other projectile, the impulsive force which produces Vega counteracts the force of gravity; in the end, this Vega is lost through friction with air, and then gravity (गुरुत्व) brings the arrow to the ground. The meaning of this "counteraction" is not clear. Is it intended that the action of gravity is suspended as long as the Vega continues? We have seen that, in the case of a body let fall, Praśastapâda expressly states that gravity (गुरुत्व) and *sanskâra* (Vega, momentum) both act in the second and following instants. Praśastapâda seems to have thought that some *sanskâras* (*e.g.* the Vega of an arrow or other projectile) suspend the action of gravity (गुरुत्वं संयोगप्रयत्नसंस्कारविरोधि, Praśastapâda, गुणग्रन्थ। वेगेन प्रतिबन्धात् अपतनं बहिःक्षिप्तस्य शरशलाकादेः निवृत्ते नोदने कर्माणि उत्तरोत्तराणि इषुसंस्कारादेवापतनादिति Praśastapâda, कर्मग्रन्थ). Other *sanskâras* (*e.g.* in the case of a falling body) coalesce with gravity to produce a single resultant motion. The later commentators (from Śrîdhara downwards) certainly interpret the Vaiśeṣika Sûtras in this sense.

Motion of a Particle in the Case of a Composition of Forces

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Any number of motions or Vegas may be impressed on a particle, but so long as these are in a uniform direction (नियतदिग्विदिष्ट) the resultant motion or Vega is in a straight line, and may be conceived as one (द्वितीयकल्पना-वैय्यर्थम्, Praśastapâda, कर्मग्रन्थ). It is only when we come to Gamana (curvilinear motion) and its causes that the question of composition assumes a real significance. In all such cases, each separate particle has only one Vega (impressed motion) in a definite direction (नियतदिगाभिमुख्य) at any given instant, but the composition of the successive motions and Vegas in the same particle produces the curvilinear motion (गमन), *e.g.* the rotation of each constituent particle of the potter's wheel. The motion of the body (अवयवी, *e.g.* the wheel) results from the combined motions of the particles (अवयवाः). If pressure or impact produces motion in an opposite direction to the Vega already impressed on the body, the original direction would be changed, as is seen in the case of rebounding (उत्पतन) after impact (अभिघात). The mortar rebounding after striking the pestle is a typical instance of such change of direction in Vega or motion. The impressed force, *e.g.* impact (अभिघात), produces a changed motion in a different direction. One view is that the original Vega (momentum) is destroyed before a new motion and a new Vega are produced by the impact. Others hold that the impact does not destroy the original Vega (momentum), but conjointly with it produces the changed motion and, through such motion, a changed Vega in a new direction. (उलूखलमुसलयोरभिघाताख्यः संयोगो-मुसलगमवेगमपेक्षमाणः मुसले उत्पतनकर्म करोति यद्यपि प्राक्तनसंस्कारो विनष्टः तथापि मुसलोलूखलयोः संयोगः पटुकर्मोत्पादकः तस्य संस्कारारम्भ-साचिव्यसमर्थो भवति अथवा प्राक्तन एव पटु संस्कारः अभिघातान् अविनश्यन् अवस्थित इति *et seq.*, Prasastapâda, कर्मग्रन्थ ।)

Typical Cases of Curvilinear Motion (Gamana)

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*Vortical Motion.*—This is due to the contact of two bodies moving in opposite directions with a like or equal Vega, *e.g.* two currents of air or water meeting from opposite directions. The change of direction is seen in the fact that water which flows downwards, or air which moves obliquely, may receive an upward motion as the result of such collision. (अथ किमिदं संमूर्च्छनं नाम। समानजयोः (तुल्यवेगयोः) वाय्वोर्विरुद्ध-दिक्क्रिययोः सन्निपातः Praśastapâda, वायुनिरूपणं). The scholiast Udayana adds: वाय्वोरितिप्रकृतत्वेन स्पर्शवत्तोरिति तु विवक्षितम्। अपां यथा द्रव्यान्तर-संमूर्च्छनाद् ऊर्द्धगमनं परस्पराभिहतनदीपयः — उदयन, द्रव्यकिरणावली. Śrîdhara notes: तिर्य्यग्गतिस्वभाव-द्रव्योदूर्द्धगतित्वात् परस्पराभिहत-जलतरङ्गोदूर्द्धगमनवत् असमानवेगयो संमूर्च्छनं न भवति एकेन अपरस्य विनयात् — Śrîdhara.)

Vibratory Motion (कम्पन, स्पन्दन)

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This will be analysed in the chapter on Acoustics.

Rotatary Motion (भ्रमण)

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Each particle of the rotating body, *e.g.* the potter's wheel (चक्र), has, at any given instant, a motion in a definite direction. The rotatory motion of the body results from the separate motions of the particles and their persistent tendencies (संस्काराः), joined with the fact of the rigid conjunction of the particles. When the rod strikes one part of the wheel, the motion in the part struck is in the first instant produced by impact (अभिघात); while the other parts move through the transmission of force due to the rigid cohesion of parts. The subsequent motions in the part struck are due to continued pressure (नोदन) and the persistent tendency (संस्कार) set up by the first motion; while the subsequent motions in the other parts are explained by their own persistent tendencies and the transmission due to rigid cohesion. When the rod is disjoined from the wheel the rotatory motion continues, being due merely to the persistent tendencies in the constituent parts and the resultant persistent tendency in the whole.

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Other varieties of curvilinear motion in bodies are to be similarly explained (*i.e.* by the composition of Vegas) (तथा चक्रादिषु अवयवानाम् पार्श्वतः प्रतिनियतदिग्देशसंयोगविभा-गोत्पत्तौ यदवयविनः संस्कारादनियतदिग्देशसंयोगविभागनिमित्तं कर्म्म तत्-भ्रमणामिति । श्चमादयो गमनविशेषाः—Praśastapâda,—कर्म्मग्रन्थ । एवं वेगात् दण्डसंयोगचक्रावयवे आद्यं कर्म्म दण्डसंयोगात् । अवयवान्तरेषु च संयुक्तसंयोगात् दण्डसंयुक्तस्य अवयवस्य उत्तरोत्तरकर्म्माणि संस्कारात् नोदनात् च । अपरेषां संस्कारात् संयुक्तसंयोगात् च दण्डविगमेतु चक्रे तदवयवेषु च संस्कारादेव केवलात् श्वमादयो गमनविशेषाः—Śrīdhara).

Section 4. Motion of Fluids

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Current motion (स्यन्दन, downward flow in a stream), upward motion (आरोहण, *e.g.* evaporation, boiling, etc.), and capillary motion (अभिसर्पण, as in plants and porous vessels) are three varieties of fluid motion which require explanation. To this may be added vortical motion (संमूर्च्छन), and स्यन्दन, wave motion, which will be noticed in the chapter on Acoustics.

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1. Current Motion (स्यन्दन).

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This is conditioned by fluidity in particles, but Śankara Miśra notes that in the downward flow of water, gravity in the fluid-particles is the efficient cause (गुरुत्वात् निमित्तकारणात् असमवायिकारणेषु—Śankara Miśra, उपस्कार). When the water is enclosed on all sides, as in a vessel, the downward flow (स्यन्दन) is counteracted. Here the fluidity does not produce motion, because, in the case of the particles in contact with the enclosing body, there is the resistance (प्रतिबन्ध) of the latter, which is transmitted to the other particles, and this counteracts the fluidity. (स्रोतोभूतानामपां स्थलाग्निसामिसर्पर्पं यत् तत् द्रवत्वात् स्यन्दनं कथं समन्तात् रोधः संयोगेन अवयविद्रवत्वम् प्रतिवद्धं उत्तरोत्तराश्वयवद्रवत्वानि संयुक्तसंयोगैः प्रतिबद्दानि — Praśastapâda, कम्मैग्रन्थ ।)

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2. Upward Motion (आरोहण, *e.g.* evaporation).

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In evaporation the fluid-particles are rarefied, and remain in a fine state of suspension; the rarefaction is due to the impulse (नोदन) or impact (अभिघात) of the heat-particles in the sun's rays, and the upward movement is due to their contact with the air under this impulse or impact. Śankara Miśra notes that in boiling there is a similar upward movement of water-particles under the impact of heat-rays. (नाड्यो वायसंयोगात् आरोहणं नोदनापीडनात् संयुक्तसंयोगात् च । Sûtras 5 and 6, Âhnika 2, Chap. 5, Vaiśeṣika Aphorism. यथा स्थालीस्था आपः क्कथ्यमानाः वायुनुन्नवह्नि-रश्मय ऊर्ध्वं नयन्ति—Śankara Miśra, उपस्कार ।)

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*N.B.*—These two Sûtras of Kanâda have been interpreted by the late Gangâdhara Kaviratna in his commentary to refer to the upward conduction of water in pipes by the pressure of air.

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The mention of the transmitted pressure (संयुक्तसंयोग) of the air seems to lend some countenance to Gangâdhara's view, and the word Nâḍi (नाड्यः) offers no difficulty, being taken in its usual sense, "pipe" (नालिका, Nalikâ), while the current interpretation does violence to the common acceptation of the word.

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3. Capillary Motion (अभिसर्पण).

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Two instances are given—the ascent of the sap in plants from the root to the stem (अभितः सर्पणं मूले निषिक्ता- नाम्पां वृक्षे—Śankara Miśra), and the penetrative diffusion of liquids in porous vessels, *e.g.* of the oil or ghee in an earthen jar (कुम्भादौ अन्तर्निहितानां तैलघृतादीनां स्यन्दनं अपनश्च). Heat-particles have a like penetrative power (दृष्टश्चानिरोधो भर्ज्जन कपालादौ तेजसः पच्यमानद्रव्यपाकसिद्धेः कलशे च निक्षिप्तानां अपां शीतस्पर्शग्रहणादनिरोधः—Jayanta, न्यायमञ्जरी, Âhnika 8, इन्द्रियाणां प्राप्यकारित्वम्).

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This is ascribed to *adṛiṣṭa*, as the cause cannot be ascertained by either perception or inference (including hypothesis) (प्रत्यक्षानुमानाभ्यां अनुपलभ्यमानकारणम्—Śrîdhara).

Section 5. Interesting Examples of Motion Ascribed to Adṛiṣṭa (Unknown Cause, Unexplained Nature, Final Cause)

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The first motions in primordial atoms at the beginning of Creation are attributed to *adṛiṣṭa*. Among movements in masses of matter so caused are noted the motion of the globe of the earth and similar other bodies (महाभूतानां भूगोलकादीनां प्रक्षोभनं चलनम्—Praśastapâda with Śrîdhara’s commentary, कर्म्मग्रन्थ). Most probably this means earthquakes, tides, etc. Âryabhata and his school would no doubt bring under this head the diurnal motion of the earth. It is interesting to note in this connection that Bhâskara refutes the Buddhist hypothesis of the earth falling perpetually *in vacuo* by arguing that the earth must remain balanced in space as there is nothing outside to attract it.

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The movement of the needle (iron in general, as Śankara Miśra notes) towards the magnet is another example of unexplained motion in matter. Cleaning and right-placing of the magnet (संमार्ज्जनम्, ऋजुस्थापनम्) are necessary (सूचीनां लौहशलाकानां, अयस्कान्ताभिमुखगमनम्, सूचीत्युपलक्षणम् अयस्कान्ताकृष्टलोहमात्रमभिप्रेतम्—Śaṅkara Miśra).

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Similarly, amber attracts grass, straw, etc. (तृणकान्तम- ण्याकृष्टानां तृणानां गमने—Śaṅkara Miśra, on Sûtra 15, Âhnika 1, Chap. V.).

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Involuntary movements of the hand under the influence of the hypnotist's *mantras* (incantations) are also attributed to *adṛiṣta*.

Section 6. Measurement of Motion.—Units of Space and Time

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The solar day was taken as a natural measure or division of time. In the Nyâya-Vaiśeṣika school the day of twenty-four hours (solar) is stated to contain 30 × 30 × 30 × 18 × 2 × 2 units of time (*kṣaṇas*). The Nyâya unit of time therefore measures 2/45 of a second. The smallest measure of time mentioned by the astrono- mers is the Truṭi, which is 1/33750 of a second.

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The natural measure of length was the cubit (Hasta), of which there were two fixed standards, the greater and the lesser cubit. The smallest measure of length mentioned in the Śilpa-śâstra (Technology) is the Paramânu, which is about 1/349525 of an inch. This is the same as the Trasareṇu of the Nyâya-Vaiśeṣika school, which stands for the thickness of the *minimum visibile* (the finest mote perceptible in the sunbeam as it comes slanting into a dark room through a chink).

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Average velocity (स्थूलगति—Bhâskara) was measured in accordance with the formula v = s/t, but no unit of velocity appears to have been fixed upon. There was no idea of acceleration, and of course no measurement of force. Mahavîrâchâryya gives formulæ for computing the space travelled over in cases of Sankalitagati (velocity with regular increment at stated intervals), but this does not amount to acceleration, as the intervals are not indefinitely small. Where the velocity is uniform, the interval of time may be of any amount (स्थूलकाल), but where the velocity is variable (प्रतिक्षणं न समागतिः—Bhâskara), an indefinitely small quantity of time (सूक्ष्मकाल) must be taken; in other words, the positions of the particle in two successive *instants* must be considered, and the velocity must be supposed to be uniform during this interval (conceived as indefinitely small, सूक्ष्म). It was in this way that Bhâskara determined the instantaneous motion of a planet (तातकालिकी गति).

Component of Velocity

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The astronomers measured the motion of a heavenly body in different directions (longitude, right ascension, etc.), and calculated separately the components of motion (गतिकलाः) in these directions, and they adopted the device of transferring such component velocities from one body to another in the computation of relative motion (*e.g.* एवं सति चाचार्येण लाघवार्थं दृष्टटोत्सङ्गभ्विन्यो गतिकला अर्क प्रक्षिप्ताः—Bhâskara, Siddhânta-Śiromaṇi, Gaṇitâdhyâya, Ghaṭisphuṭî-prakaraṇa).

Section 7. Notion of Three Axes

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Motion, we have seen, was defined as the change of position of a particle in space. To conceive position in space, Vâchaspati takes three axes, and the position in space of one particle relatively to another may be indicated by distances measured along these three axes. This remarkable analysis (*circa* 842 A.D. : वस्वङ्कवसुवत्सरे, Vâchaspati, Nyâyasûchînibandha) anticipates in a rudimentary manner the foundations of solid (co-ordinate) geometry, eight centuries before Descartes (*vide* p. 118, *supra*).

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*The Principle of the Differential Calculus applied to the Computation of Motion (Variable Motion).*

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Bhâskara (1150 A.D.), in computing what he calls the “instantaneous” motion (तात्कालिकी गति) of a planet, compares its successive positions in two successive instants, and regards the motion as constant during the interval, which he conceives to be indefinitely small (त्रुटिकाल). This is equivalent to the determination of the differential of the planet’s longitude, and the process bears “a strong analogy” (to quote the words of Mr. Spottiswoode, the Astronomer Royal) “to the corresponding process in modern mathematical astronomy.” I have elsewhere shown that Bhâskara’s process was not merely analogous to, but virtually identical with, that of the Differential Calculus, Mr. Spottiswoode’s cautious reservation having been due to his want of acquaintance with the original and the insufficiency of the materials placed before him (*vide* pp. 78–79 *supra*).

Section 8. Relative Motion

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The phenomenon is noticed among the hallucinations of sense (नाव्यारुढाश्व गच्छतः पर्वतादीनि विजानन्ति अभ्रेश भ्रमत्तश्च तान्—Kumârila, Śloka-Vârtika, p. 520). Astronomers like Âryabhata and Lalla, who believed in the diurnal revolution of the earth from the west to the east, explained the apparent revolution of the starry heavens in the opposite direction by the principle of relative motion.

Section 9. Serial Motion

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Several Santânas (series) of motions are incidentally noticed, *e.g.* vibration (स्पन्दन, कम्पसन्तान), wave motion (वीचितरङ्ग), current motion (स्यन्दन).

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In an interesting passage, Charaka notes three instances of serial motion, viz. those of water, sound, and light (जलसन्तान, शब्दसन्तान and अर्चिःसन्तान), to which he compares the course (सन्तान) of chyle (or chyle-blood) in the Dhamanîs (veins) and other ducts of the body.

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Dalvana thinks that downward, oblique, and upward currents of chyle are respectively intended by the three illustrations; but Chakrapâṇi points out that the Santâna (wave) of sound travels in all directions (the same is of course true also of light), and that differences in speed (and not in direction) are here meant. In other words, a Santâna of sound travels more rapidly than that of water, and less rapidly than one of light (अर्चिःसन्तान), and Charaka's meaning is that the metabolic course may complete its circuit with greater or less speed. Whether, in this passage, the three Santânas are viewed as waves or currents, is not specified; but the difference between a wave (वीचि) and a current (स्यन्दन) was well known.

c5.s9.5

A current of water (स्यन्दन, downward flow) consists of particles moving in an uninterrupted series under the action of gravity and fluidity (गुरुत्व and द्रवत्व, Śankara Miśra). A wave (वीचितरङ्ग), on the other hand, is constituted by the transmission of vibratory motion (स्पन्दन) in the water-particles (*e.g.* Jayanta, सततष्टनव-योविन्दुसन्दोहस्पन्दनक्रमात्—Nyâya-Manjarî, Âhnika 2).

c5.s9.6

A ray of light, on the other hand, was supposed to imply the rectilinear propagation of indefinitely minute corpuscles in all directions, with inconceivable velocity, and a sort of conical dispersion (अचिन्त्योहि वेगातिशयः तेजः प्रसर्पति पृथ्वग्रत्वात्—Udyotakara, Vâchaspati).

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