משנה: הַנּוֹטֵעַ יֶרֶק בְּכֶרֶם אוֹ הַמְּקַייֵם הֲרֵי זֶה מְקַדֵּשׁ אַרְבָּעִים וְחָמֵשׁ גְּפָנִים. אֵימָתַי בִּזְמַן שֶׁהֵן נְטוּעוֹת אַרְבַּע עַל אַרְבַּע אוֹ חָמֵשׁ עַל חָמֵשׁ. הָיוּ נְטוּעוֹת עַל שֵׁשׁ שֵׁשׁ אוֹ עַל שֶׁבַע שֶׁבַע הֲרֵי זֶה מְקַדֵּשׁ שֵׁשׁ עֶשְׂרֵה אָמָּה לְכָל־רוּחַ עֲגוֹלוֹת וְלֹא מְרוּבָּעוֹת. MISHNAH: He who plants vegetables in a vineyard or who lets them grow60If the edible vegetable grew from wind-borne seeds. sanctifies 45 vines. When is this? In case they are planted four four or five five apart. If they were planted six six or seven seven apart, he sanctifies sixteen cubits in every direction, circular but not square61The language of this Mishnah is so involved because of the special rule for a vineyard planted in spaces of five cubits. In general, vegetables can be planted in a vineyard only in a bald spot; according to Mishnah 4:1 that means at a distance of 16 cubits from the next vine. The rule of the Mishnah for vines planted more than 5 cubits apart is just that: In the interior of the circle of radius 16 and centered at the vegetable there may not be any vine.
The application of that principle to vines planted 4 apart leads to 45 sanctified vines. One assumes that the vegetable is sown closely around a vine. One takes the stern of that vine as center of a Cartesian coordinate system and takes the x-axis in the direction of the rows of vines, the y- axis perpendicular. The unit of measurement is the cubit. Points with coordinates (4m,4n) with integer m,n represent vines. According to Pythagoras, the distance from the origin is <16 if and only if m2+n2< 16. This means that for m = 0, ±1, ±2, n = 0, ±1, ±2, ±3, and for m = ±3, n = 0, ±1, ±2. In all, there are 5·7 + 2·5 = 45 admissible couples (m,n). This is explained in the Halakhah. There the language is that all acceptable couples are m,n = 0, ±1, ±2, ±3 except for couples (±3,±3). The case of vines planted 5 apart is not treated; one has to assume that one excludes the vines at points (5m,5n) for m2+n2< 25 because vineyards planted 5 apart look similar to those planted 4 apart. (For an opposing opinion in the Tosephta cf. Tosefta ki-fšuṭah p. 629.).
הלכה: רִבִּי יוֹסֵי בֶּן חֲנִינָא אָמַר וְהוּא שֶׁזָּרַע כְנֶגֶד הָאֶמְצָעִית. רִבִּי אָבִין בְּשֵׁם שְׁמוּאֵל וְהוּא שֶׁתְּהֵא הָאֶמְצָעִי עֲגוֹלָה יֶרֶק. הֵיךְ עֲבִידָא. שׁוּבָע שׁוּרִין מִן שׁוּבָע שׁוּרִין. צֵא מֵהֶן אַרְבַּע גְּפָנִים לְאַרְבַּע זְוִיּוֹת הַכֶּרֶם נִשְׁתַּייְרוּ שָׁם אַרְבָּעִים וְחָמֵשׁ גְּפָנִים. הָדָא הִיא דְתַנִּינָן הֲרֵי [הוּא] מְקַדֵּשׁ אַרְבָּעִים וְחָמֵשׁ גְּפָנִים. HALAKHAH: Rebbi Yose ben Ḥanina said, only if he sowed near the middle vine. Rebbi Abin in the name of Samuel: Only if the middle vine is surrounded by vegetables. How is it computed? Seven rows by seven rows. Subtract from them four vines at the four corners; there are 45 vines remaining. That is what we stated: “He sanctifies 45 vines”62All this was explained in Note.
אֵימָתַי בִּזְמַן שֶׁהֵן נְטוּעוֹת אַרְבַּע עַל אַרְבַּע אוֹ חָמֵשׁ עִל חָמֵשׁ. הָדָא מְסַייְעָא לְרִבִּי זְעִירָא דְּרִבִּי זְעִירָא אָמַר שְׁמוֹנֶה חוּץ מִמְּקוֹם כָּרְתּוֹן. הָיוּ נְטוּעוֹת שֵׁשׁ עַל שֵׁשׁ אוֹ שֶׁבַע עַל שֶׁבַע הָדָא מְסַייְעָא לְרִבִּי לָֽעְזָר דְּרִבִּי לָֽעְזָר אָמַר מֵאַרְבַּע אַמּוֹת וְעַד שְׁמוֹנֶה אָסוּר וּמְקַדֵּשׁ. מִשְּׁמוֹנֶה וְעַד שֵׁשׁ עֶשְׂרֵה אָסוּר וְאֵינוֹ מְקַדֵּשׁ. “When is this? In case they are planted four four or five five apart.” This supports Rebbi Zeïra, since Rebbi Eleazar63This has to be “R. Zeïra”, cf. Chapter 4, Note 122. If the trunks were counted in the distances, the four vines at distance 16 from the center on the axes would also be forbidden, for a total of 49, not 45. said: Eight except for the place of the trunks. “If they were planted six six or seven seven apart”, that supports Rebbi Eleazar, since Rebbi Eleazar64In Chapter 4, Halakhah 8 (Note 125), this also is a statement of R. Zeïra. The argument here is that “eight eight” is not mentioned, which implies that vegetables in a vineyard planted eight cubits apart do not “sanctify”. said: From four cubits to eight, it is forbidden and “sanctifies”, from eight to sixteen it is forbidden but does not “sanctify”.