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Fig.: बहुव्रीहिः पुरुषः
Jodhpur = जोधपुर
(Image source: Details)
बहुव्रीहिः = बहवो व्रीहयो यस्य सः = "one who possesses much rice"
In contrast to a Tatpuruṣa, a Bahuvrīhi denotes something other than what is expressed by its members: while बालपुत्रः as a Tatpuruṣa resolves to "a young son," i.e., something expressed by the final member of the compound (पुत्र), बालपुत्रः as a Bahuvrīhi resolves to "one whose son is young," i.e., the denoted entity is neither the son (पुत्र) nor necessarily young (बाल), but a person distinct from the young son, namely his father.
Possessive compounds are initially always adjectives, which can however be substantivized. Therefore, the grammatical gender of a Bahuvrīhi depends on what the Bahuvrīhi refers to, and not on the gender of the final compound member.
Schema for resolving a Bahuvrīhi:
1. First member usually in the Nominative (प्रथमा) Singular, Dual or Plural -- 2. Member in the Nominative Singular, Dual or Plural -- Relative pronoun in a case other than the Nominative (usually Genitive - षष्ठी) and in gender and number of the total Bahuvrīhi -- Demonstrative pronoun in number, case and gender of the total Bahuvrīhi.
Examples:
गतपापः = गतं पापं यस्य सः = "one whose evil has gone" = "one who is free from misdeeds"
Acc. sg. गतपापम् = गतं पापं यस्य तम्
Inst. sg. गतपापेन = गतं पापं यस्य तेन
Gen. sg. गतपापस्य = गतं पापं यस्य तस्य
Nom. pl. गतपापाः = गतानि पापानि yeषां ते
etc.
अस्तमोहा = अस्तो मोहो यया सा = "a (woman) who has cast off delusion"
प्राप्तोदको ग्रामः = प्राप्तमुदकं यं s ग्रामः = "a village to which the water has reached" = "a village threatened by flooding"
पुण्यवत्पुत्रः as Bahuvrīhi = पुण्यवान्पुत्रो यस्य सः, or: पुण्यवन्तः पुत्रा यस्य सः = "one whose son possesses merit" or: "one whose sons possess merit"
The relationship between the first member and the second member in a Bahuvrīhi can be:
As with the Tatpuruṣa, the specified member usually stands in second place.
Traditionally, one distinguishes:
समानाधिकरणबहुव्रीहिः : in विग्रहवाक्य the first member and the second member are in the same case
व्यधिकरणबहुव्रीहिः : in विग्रहवाक्य the first member and the second member are in different cases
Schema:
Adjective -- Noun
Example:
गुणवत्पुत्रो ब्राह्मणः = गुणवान्पुत्रो यस्य स ब्राह्मणः = "a Brāhmaṇa whose son / sons possesses / possess good qualities"
Attributive determination by a PPP (Past Participle), which is rare in Tatpuruṣa, occurs particularly frequently. The relative pronoun then usually stands in the Instrumental (तृतीया) upon resolution (rarely in the Genitive / षष्ठी).
Rule of thumb:
Example:
कृतफल ३ = कृतं फलं yeन सः । यया सा । yeन तत् = "one / a / an, who / which has produced a result, someone / something effective"
Schema:
noun -- noun
Examples:
शूरपुत्रो नरः = शूरा एव पुत्रा यस्य स नरः = "a man whose sons are heroes"
तदन्त ३ = सो ऽन्तो यस्य सः । यस्याः सा । यस्य तत् = "who / which, whose end is this" = "ending with that"
Very important are appositional Bahuvrīhis with the second members आदि । आदिक । आद्य "first, beginning", less frequently प्रभृति "beginning". Such compounds correspond to "etc.":
Example:
देवा इन्द्रादयः = इन्द्र आदिर्येषां ते देवाः = "the gods whose beginning is Indra" = "Indra etc." = "Indra and the other gods"
To express "only", one can use bahuvrīhis with मात्रा f. "measure, limit" as the second member:
Example:
शब्दमात्रम् = शब्दो मात्रा यस्य तत् = "that whose measure is a word" = "only one word"
मात्र after a PPP is to be translated as "as soon as":
Example:
जातमात्रं शत्रुं घ्नन्ति = जातं मात्रा यस्य तं शत्रुं घ्नन्ति = "they slay an enemy whose measure is arising" = "they slay the enemy as soon as it has arisen"

Fig.: देवा यीश्वादयः
Jesus and the other gods, Ahmedabad
(Image source: Details)
i.e., Bahuvrīhis whose first member represents a case other than the nominative (प्रथमा).
Example:
देवरूपा = देवस्यैव रूपं यस्याः सा = "a (woman) whose form is that of a god" "a woman with divine appearance"
Exception to the general rule regarding the order of compound members:
The case-marked member stands in second place when it denotes a body part (esp. the hand):
Example:
दण्डहस्तः = दण्डो हस्ते (Loc. sg.) यस्य सः = "one in whose hand a staff is located" = "one who carries a staff in the hand."

Fig.: इन्द्रो वज्रपाणिः
Siemreap (ក្រុងសៀមរាប), Cambodia, 9th cent. CE
(Image source: Details)
An adverb, a preposition, a prefix, or another indeclinable stands in the first member; such compounds are generally नित्यसमास.
Examples:
अपुत्रो नरः = पुत्रो यस्य नास्ति स नरः = "a man who has no son" = "a sonless man"
दुर्बल ३ = "someone / something whose strength is bad" = "powerless, weak"
सह "with" as the first member of a Bahuvrīhi is often replaced by स.
Example:
सपुत्रः = सहपुत्रः = पुत्रेण सहितः । पुत्रेण सह् = "with a son", "accompanied by a son"

Fig.: सपुत्रा
Madhya Pradesh
(Image source: Details)
The final member of a Bahuvrīhi is inflected as an adjective with multiple genders, regardless of its original gender. Thus, if in a compound the final member has a different gender than its original one, it must be a Bahuvrīhi if it is not a Samāhāradvanva (Neuter singular).
When adjectivizing,
The examples are given here for later study; for some, the necessary grammar and vocabulary have not yet been discussed

Fig.: कृतकटा
Chennai = சென்னை
(Image source: Details)
Although every bahuvrīhi can be resolved into a relative clause, not every relative clause can be replaced by a bahuvrīhi. Hermann Georg Jacobi (1850 - 1937), who examined the relationship between the two in greater detail (Compositum und Nebensatz, 1897), writes:
"When translating from German or another modern language into Sanskrit, not all subordinate clauses may be rendered by compounds; only the embellishing and descriptive ones are suitable. Those, however, which contain a conceptually necessary or important determination also appear in Sanskrit as relative clauses."

Fig.: Hermann Jacobi
(Image source: Details)
That may be true. However, the Indians then view many things as embellishing and descriptive that we would consider "conceptually necessary or important."
वा : or
आश्रम m., n.: hermitage, stage of life, phase of life (namely as ब्रह्मचरिन्, गृहस्थ, वनप्रस्थ and possibly as सन्न्यासिन् ; see Basham, Wonder p. 159f.)

Fig.: आश्रमः
Rishikesh = ऋषिकेश. "This is actually part of the Ashram in Rishikesh where the Beatles stayed. This specific part was not there when they were there, but it does look cool."
(Image source: Details)
कर ३ f. करी । करा : doing, making, causing
कर m.: hand (to कृ 8)
कर m.: payment, tribute, tax (not to कृ, but probably a loanword from Tamil - தமிழ்)
क्रिया f.: action, sacred action, sacrificial act, ceremony (to कृ 8)
अधि preverb: over, on, up-, with reference to
गम् +अधि 1 P अधिगच्छति : encounter, reach, attain
तनूकृ 8 U तनूकरोति : diminish, weaken
दायक ३ f.: दायिका : giving, bestowing
नृप m.: "protector of men" = king
प्रणिधान n.: application, effort, attention towards, helpfulness, reflection, vow
बाधना f.: distress, plague, torment
भार्या f.: "one to be maintained" = wife
भावना f.: meditative unfolding (to भू causative)
मही f.: earth, ground and soil
लक्षण n.: feature, sign, attribute
विप्र m.: "trembler" = poet, singer, priest, Brahmin
विषय m.: area, region, object, sense-object
अपवर्ग m.: end, liberation
नि preverb: downwards, down, into, backwards
वृत् + नि 1 Ā निवर्तते : turn around, return
सद् 1 P सीदति (!) Pass. सद्यते PPP सन्न : sit, settle down
सद् + प्र 1 P प्रसीदति : to sit, be seated (in a figurative sense) = become composed, calm, cheerful; to be gracious towards someone (genitive षष्ठी)
समाधि m.: inner concentration, highest attention, meditative "absorption"
स्वाध्याय m.: "self-study", recitation (especially of the Veda), study of the Vedas
परलौकिक ३ : pertaining to the other world, beyond
तनु ३ : slender
मध्य ३ : middle; n. center
पृथु ३ (पृथ्वी) : wide, broad, large
श्रोणि । श्रोणी f.: hip
रक्त ३ : colored, red
ओष्ठ m.: lip
असित ३ : dark, black
ईक्ष् 1 Ā ईक्षते Pass. ईक्ष्यते PPP ईक्षित : to see
नम् 1 P नमति Pass. नम्यते PPP नत : to bend
उद् Preverb: up, upward, out, from, ex-
नाभि f.: navel
वपुस् n.: beauty, form body (declension see later)
स्त्री f.: woman
स्तन m.: breast
दरैद्र ३ : poor, weak
ऋध् 5 P ऋध्नोति Pass. ऋध्यते PPP ऋद्ध : to thrive
ऋध् + सम् : to thrive; PPP: successful, rich
विचित्र ३ : colorful, diverse, beautiful, wonderful, strange
विधि m.(!): arrangement, law, regulation; creation, fate
चेष्ट् 1 Ā चेष्टते Pass. चेष्ट्यते PPP चेष्टित : to stir, move
Resolve the following compounds as Bahuvrīhi and/or Dvandva and/or Tatpuruṣa in all ways that seem possible to you in Sanskrit (exception: compounds with adverbial first member). Translate these differently resolved compounds into German, indicating the gender, case, and number of the entire compound.
A) Translate and resolve the compounds into Sanskrit:
इन्द्रशत्र्वनार्या देवेन्द्रेण जीयन्ते ॥१॥ शूरबलक्षत्रिययोधः शूरपुत्रमिच्छति ॥२॥ सुदुर्गममार्गेण स्वर्गं गम्यते । सुगमस्तु नरकमार्गः ॥३॥ मृतपुत्रब्राह्मणी रोदिति ॥४॥ वीतमोहब्राह्मणः सम्पन्नरूपामपि शूद्रां न लुभ्यति ॥५॥ सुनीतिपुत्रः प्राप्तमतिदर्शनसाधुं गच्छति ॥६॥ प्राप्तप्रभावक्षत्रिया दृष्टमात्राञ्छत्रून्घ्नन्ति ॥७॥ जितशत्रुयोधाः शत्रुजितान्मुञ्चन्ति ॥८॥ कृतोपनयनबालः शिवादिदेवपूजां करोति ॥९॥ बुद्धगता दुःखादिसत्यानि शृण्वन्ति ॥१०॥
B) Translate into Sanskrit using compounds:
A Kṣatriya who does not hold the staff in his hand does not protect the people.
Kālidāsa and the other poets are the teachers in Sanskrit.
A Kṣatriya has his livelihood through weapons.
Śūdra women also have non-violence, truthfulness, purity, not grumbling, not being malicious, and forbearance as their dharma.
मैत्रीकरुणामुदितोपेक्षाणां सुखदुःखपुण्यापुण्यविषयाणां भावनतश्चित्तप्रसादनम् ॥योगसूत्र १.३३॥
Explanation: भाव्नातस् = भावना + suffix -tas, which has an ablative meaning. Translate: "because of ..." or similar.
तपःस्वाध्यायेश्वरप्रणिधानानि क्रियायोगः ॥योगसूत्र २.१॥ समाधिभावनार्थः क्लेशतनूकरणार्थश्च ॥योगसूत्र २.२॥
Explanation: तपस् n. (declension later): glow, heat, torment; ascetic fervor, austerity

Fig.: तपस्
Buddha as an ascetic before his liberating insight, Gandhara, 2nd/3rd century.
(Image source: Details)
त्रिविधदुःखात्यन्तनिवृत्तिरत्यनपुरुषार्थः ॥सांख्यसूत्र १.१॥ (For the Sāṃkhya system, see Basham, Wonder pp. 324f.)
Explanation: त्रिविध ३ : "threefold"
Definition of liberation according to the Nyāya system:
बाधनालक्षणं दुःखम् ॥न्यायसूत्र १.२१॥ तदत्यन्तविमोक्षो ऽपवर्गः ॥न्यायसूत्र १.२२॥
Utterance of the Materialists according to the सर्वदर्शनसंग्रह :
न स्वर्गो नापवर्गो वा नैवात्मा पारलौकिकः । नैव वर्णाश्रमादीnaं क्रियाश्च फलदायिकाः ॥
Explanation: अत्मा = Nom.sg. masc. to आत्मन् m. "Self, soul; the Absolute insofar as it is realized in an individual"
A सुभाषितम् :
देवानां करदा विप्रा विप्राणां करदा नृपाः । नृपाणां करदा लोका लोकानां करदा मही ॥
Explanation: -da at the end of a compound: "giving"

Fig.: लोकानां करदा मही
Karnataka
(Image source: Details)
On matters of property:
भार्या पुत्रश्च दासश्च त्रय एवाधनाः स्मृताः । यत्ते समधिगच्छन्ति यस्य ते तस्य तद्धनम् ॥मनुस्मृति ८.४१६॥
Explanation: त्रयस् = Nom. masc. pl. to त्रि "three"
A सुभाषितम् on female beauty:
तनुमध्यं पृथुश्रोणि रक्तौष्ठमसितेक्षणम् । नतनाभि वपुः स्त्रीणां कं न हन्त्युन्नतस्तनम् ॥
Explanation: all forms except कं and स्त्रीणाम् are Nom. sg. neuter and refer to वपुस्.

Fig.: तनुमध्यं पृथुश्रोणि
Sanchi = सांची
(Image source: Details)
Another सुभाषितम् :
सन्ति पुत्राः सुबहवो दरिद्राणामनिच्छताम् । नास्ति पुत्रः समृद्धानां विचित्रं विधिचेष्टितम् ॥
Explanation: इच्छताम् = gen. pl. Pres. Part P to इष् : desiring