In Tzav, we ended with a 7 day inauguration of the Mishkan as well as the initiaion of Aaron and his sons into being kohanim. This parsha (the 26th of the weekly portions) begins with Shemini (8), the next day.
Moses gives instructions to Aaron and his sons about the offerings and the atonement for sins that they or the people have committed.
Two of Aaron's sons, Nadav and Abihu, bring fire offerings of a 'foreign' nature. Because these were not commanded by God, they are punished by death.
Aaron responds with silence. Moses tells Aaron and his other sons (Eleazar and Ithamar) not to mourn for them.
Moses 'calls out' Aaron on a point of law about offerings and concedes that Aaron has done the correct thing.
God tells Moses and Aaron which foods are permitted for eating and which are forbidden to the people of Israel.
Land animals need to have split hooves and chew their cud. Fish must have fins and scales. Birds of prey are not permitted (a list is given along with a list of kosher insects).
Details about the laws of ritual purity and the differences between the impure and the pure.
וַיְהִי֙ בַּיּ֣וֹם הַשְּׁמִינִ֔י קָרָ֣א מֹשֶׁ֔ה לְאַהֲרֹ֖ן וּלְבָנָ֑יו וּלְזִקְנֵ֖י יִשְׂרָאֵֽל׃
(ג) וְאֶל־בְּנֵ֥י יִשְׂרָאֵ֖ל תְּדַבֵּ֣ר לֵאמֹ֑ר קְח֤וּ שְׂעִיר־עִזִּים֙ לְחַטָּ֔את וְעֵ֨גֶל וָכֶ֧בֶשׂ בְּנֵי־שָׁנָ֛ה תְּמִימִ֖ם לְעֹלָֽה׃ (ד) וְשׁ֨וֹר וָאַ֜יִל לִשְׁלָמִ֗ים לִזְבֹּ֙חַ֙ לִפְנֵ֣י יְהֹוָ֔ה וּמִנְחָ֖ה בְּלוּלָ֣ה בַשָּׁ֑מֶן כִּ֣י הַיּ֔וֹם יְהֹוָ֖ה נִרְאָ֥ה אֲלֵיכֶֽם׃
(3) And speak to the Israelites, saying: Take a he-goat for a sin offering; a calf and a lamb, yearlings without blemish, for a burnt offering; (4) and an ox and a ram for an offering of well-being to sacrifice before יהוה; and a meal offering with oil mixed in. For today יהוה will appear to you.”
וַיֹּ֣אמֶר מֹשֶׁ֔ה זֶ֧ה הַדָּבָ֛ר אֲשֶׁר־צִוָּ֥ה יְהֹוָ֖ה תַּעֲשׂ֑וּ וְיֵרָ֥א אֲלֵיכֶ֖ם כְּב֥וֹד יְהֹוָֽה׃
Moses said: “This is what יהוה has commanded that you do, that the Presence of יהוה may appear to you.”
In kabbalistic teachings, the number seven symbolizes perfection – perfection that is achievable via natural means – while eight symbolizes that which is beyond nature and its (inherently limited) perfection.....Eight, on the other hand, is symbolic of an entity that is one step above the natural order, higher than nature and its limitations.
Eliezer Posner, Chabad.org
(יב) וּבֶן־שְׁמֹנַ֣ת יָמִ֗ים יִמּ֥וֹל לָכֶ֛ם כׇּל־זָכָ֖ר לְדֹרֹתֵיכֶ֑ם יְלִ֣יד בָּ֔יִת ...
(12) And throughout the generations, every male among you shall be circumcised at the age of eight days.
(א) לַמְנַצֵּ֥חַ עַֽל־הַשְּׁמִינִ֗ית מִזְמ֥וֹר לְדָוִֽד׃
Thus, while everything natural and normative in life and in Judaism is connected to the number seven, everything transcendent in Jewish life is connected to the number eight. ~www.chabad.org

Rabbi Michael Munk, The Wisdom in the Hebrew Alphabet
Rabbi Yehuda continues: And the harp that will be played in the days of the Messiah will have eight strings, as it is stated: “For the Leader, on the eighth: A Psalm of David” (Psalms 12:1). This indicates that the psalms that will be recited in the time of the Messiah, son of David, will be played on the eighth string that will be added to the harp.
After citing several verses where vayhi portends grief, the Gemara mentions a number of verses that seem to indicate otherwise. But isn’t it written: “And it came to pass [vayhi] on the eighth day” (Leviticus 9:1), which was the day of the dedication of the Tabernacle? And it is taught in a baraita with regard to that day: On that day there was joy before the Holy One, Blessed be He, similar to the joy that existed on the day on which the heavens and earth were created. The Gemara cites a verbal analogy in support of this statement. It is written here, with regard to the dedication of the Tabernacle: “And it came to pass [vayhi] on the eighth day,” and it is written there, in the Creation story: “And it was [vayhi] evening, and it was morning, one day” (Genesis 1:5). This indicates that there was joy on the eighth day, when the Tabernacle was dedicated, similar to the joy that existed on the day the world was created. Apparently, the term vayhi is not necessarily a portent of grief.
Then Moses said to Aaron: “Come forward to the altar and sacrifice your sin offering and your burnt offering, making expiation for yourself and for the people; and sacrifice the people’s offering and make expiation for them, as יהוה has commanded.” Aaron came forward to the altar and slaughtered his calf of sin offering.
וַיַּקְרֵ֕ב אֵ֖ת קׇרְבַּ֣ן הָעָ֑ם וַיִּקַּ֞ח אֶת־שְׂעִ֤יר הַֽחַטָּאת֙ אֲשֶׁ֣ר לָעָ֔ם וַיִּשְׁחָטֵ֥הוּ וַֽיְחַטְּאֵ֖הוּ כָּרִאשֽׁוֹן׃ וַיַּקְרֵ֖ב אֶת־הָעֹלָ֑ה וַֽיַּעֲשֶׂ֖הָ כַּמִּשְׁפָּֽט׃
Next he brought forward the people’s offering. He took the goat for the people’s sin offering, and slaughtered it, and presented it as a sin offering like the previous one. He brought forward the burnt offering and sacrificed it according to regulation.
וַיִּשָּׂ֨א אַהֲרֹ֧ן אֶת־יָדָ֛ו אֶל־הָעָ֖ם וַֽיְבָרְכֵ֑ם וַיֵּ֗רֶד מֵעֲשֹׂ֧ת הַֽחַטָּ֛את וְהָעֹלָ֖ה וְהַשְּׁלָמִֽים׃ וַיָּבֹ֨א מֹשֶׁ֤ה וְאַהֲרֹן֙ אֶל־אֹ֣הֶל מוֹעֵ֔ד וַיֵּ֣צְא֔וּ וַֽיְבָרְכ֖וּ אֶת־הָעָ֑ם וַיֵּרָ֥א כְבוֹד־יְהֹוָ֖ה אֶל־כׇּל־הָעָֽם׃
Aaron lifted his hands toward the people and blessed them; and he stepped down after offering the sin offering, the burnt offering, and the offering of well-being. Moses and Aaron then went inside the Tent of Meeting. When they came out, they blessed the people; and the Presence of יהוה appeared to all the people.
וַתֵּ֤צֵא אֵשׁ֙ מִלִּפְנֵ֣י יְהֹוָ֔ה וַתֹּ֙אכַל֙ עַל־הַמִּזְבֵּ֔חַ אֶת־הָעֹלָ֖ה וְאֶת־הַחֲלָבִ֑ים וַיַּ֤רְא כׇּל־הָעָם֙ וַיָּרֹ֔נּוּ וַֽיִּפְּל֖וּ עַל־פְּנֵיהֶֽם׃
Fire came forth from before יהוה and consumed the burnt offering and the fat parts on the altar. And all the people saw, and shouted, and fell on their faces.
וַיִּקְח֣וּ בְנֵֽי־אַ֠הֲרֹ֠ן נָדָ֨ב וַאֲבִיה֜וּא אִ֣ישׁ מַחְתָּת֗וֹ וַיִּתְּנ֤וּ בָהֵן֙ אֵ֔שׁ וַיָּשִׂ֥ימוּ עָלֶ֖יהָ קְטֹ֑רֶת וַיַּקְרִ֜יבוּ לִפְנֵ֤י יְהֹוָה֙ אֵ֣שׁ זָרָ֔ה אֲשֶׁ֧ר לֹ֦א צִוָּ֖ה אֹתָֽם׃
Now Aaron’s sons Nadab and Abihu each took his fire pan, put fire in it, and laid incense on it; and they offered before יהוה alien fire, which had not been enjoined [read commanded] upon them.
וַתֵּ֥צֵא אֵ֛שׁ מִלִּפְנֵ֥י יְהֹוָ֖ה וַתֹּ֣אכַל אוֹתָ֑ם וַיָּמֻ֖תוּ לִפְנֵ֥י יְהֹוָֽה׃
And fire came forth from יהוה and consumed them; thus they died at the instance of יהוה.
Then Moses said to Aaron, “This is what יהוה meant by saying:
Through those near to Me I show Myself holy,
And gain glory before all the people.”
And Aaron was silent.
And so do not go outside the entrance of the Tent of Meeting, lest you die, for יהוה’s anointing oil is upon you.” And they did as Moses had bidden.
And Moses said to Aaron and to his sons Eleazar and Ithamar, “Do not bare your heads and do not rend your clothes, lest you die and anger strike the whole community. But your kin, all the house of Israel, shall bewail the burning that יהוה has wrought.
וַיְדַבֵּ֣ר יְהֹוָ֔ה אֶֽל־אַהֲרֹ֖ן לֵאמֹֽר׃ יַ֣יִן וְשֵׁכָ֞ר אַל־תֵּ֣שְׁתְּ ׀ אַתָּ֣ה ׀ וּבָנֶ֣יךָ אִתָּ֗ךְ בְּבֹאֲכֶ֛ם אֶל־אֹ֥הֶל מוֹעֵ֖ד וְלֹ֣א תָמֻ֑תוּ חֻקַּ֥ת עוֹלָ֖ם לְדֹרֹתֵיכֶֽם׃ וּֽלְהַבְדִּ֔יל בֵּ֥ין הַקֹּ֖דֶשׁ וּבֵ֣ין הַחֹ֑ל וּבֵ֥ין הַטָּמֵ֖א וּבֵ֥ין הַטָּהֽוֹר׃ וּלְהוֹרֹ֖ת אֶת־בְּנֵ֣י יִשְׂרָאֵ֑ל אֵ֚ת כׇּל־הַ֣חֻקִּ֔ים אֲשֶׁ֨ר דִּבֶּ֧ר יְהֹוָ֛ה אֲלֵיהֶ֖ם בְּיַד־מֹשֶֽׁה׃ {פ}
And יהוה spoke to Aaron, saying: Drink no wine or other intoxicant, you or your sons, when you enter the Tent of Meeting, that you may not die. This is a law for all time throughout the ages, for you must distinguish between the sacred and the profane, and between the impure and the pure; and you must teach the Israelites all the laws which יהוה has imparted to them through Moses.
(טז) וְאֵ֣ת ׀ שְׂעִ֣יר הַֽחַטָּ֗את דָּרֹ֥שׁ דָּרַ֛שׁ מֹשֶׁ֖ה וְהִנֵּ֣ה שֹׂרָ֑ף וַ֠יִּקְצֹ֠ף עַל־אֶלְעָזָ֤ר וְעַל־אִֽיתָמָר֙ בְּנֵ֣י אַהֲרֹ֔ן הַנּוֹתָרִ֖ם לֵאמֹֽר׃
(16) Then Moses inquired about the goat of sin offering, and it had already been burned! He was angry with Eleazar and Ithamar, Aaron’s remaining sons, and said,
(17) “Why did you not eat the sin offering in the sacred area? For it is most holy, and it is what was given to you to remove the guilt of the community and to make expiation for them before יהוה. (18) Since its blood was not brought inside the sanctuary,*brought inside the sanctuary As is done in the case of the most solemn offerings; see 4.3–21; 16.11–17. you should certainly have eaten it in the sanctuary, as I commanded.” (19) And Aaron spoke to Moses, “See, this day they brought their sin offering and their burnt offering before יהוה, and such things have befallen me! Had I eaten sin offering today, would יהוה have approved?” (20) And when Moses heard this, he approved.
(ג) שְׁתֵּי קְצָווֹת הָרְחוֹקוֹת זוֹ מִזּוֹ שֶׁבְּכָל דֵּעָה וְדֵעָה אֵינָן דֶּרֶךְ טוֹבָה וְאֵין רָאוּי לוֹ לָאָדָם לָלֶכֶת בָּהֶן וְלֹא לְלַמְּדָן לְעַצְמוֹ. וְאִם מָצָא טִבְעוֹ נוֹטֶה לְאַחַת מֵהֶן אוֹ מוּכָן לְאַחַת מֵהֶן אוֹ שֶׁכְּבָר לָמֵד אַחַת מֵהֶן וְנָהַג בָּהּ יַחֲזִיר עַצְמוֹ לַמּוּטָב וְיֵלֵךְ בְּדֶרֶךְ הַטּוֹבִים וְהִיא הַדֶּרֶךְ הַיְשָׁרָה:
Each middah has two extremities. It is not a good path nor appropriate to go according to these extremes. … If you find that your nature is leaning towards one of them, or that your are 'prepared' for one of them, or you have learned and become accustomed to one of them - he should return himself to the 'good path' and go in the good way which is the straight path.
הַכַּעַס מֵבִיא לִידֵי טָעוּת; מִי לָנוּ גָּדוֹל מִמֹּשֶׁה רַבֵּנוּ עָלָיו הַשָּׁלוֹם, שֶׁכָּעַס בִּשְׁלוֹשָׁה מְקוֹמוֹת וּבָא לִכְלַל טָעוּת: כְּמוֹ שֶׁנֶּאֱמַר (ויקרא י טז): ״וַיִּקְצֹף עַל אֶלְעָזָר וְעַל אִיתָמָר״; וּכְתִיב (שם פסוק יז): ״מַדּוּעַ לֹא אֲכַלְתֶּם אֶת הַחַטָּאת בִּמְקוֹם הַקֹּדֶשׁ״. וְנֶאֱמַר (במדבר כ י): ״שִׁמְעוּ נָא הַמּוֹרִים״; וּכְתִיב (שם פסוק יא): ״וַיַּךְ אֶת הַסֶּלַע״. וּכְתִיב: ״וַיִּקְצֹף מֹשֶׁה עַל פְּקוּדֵי הֶחָיִל״ (שם לא יד); וּכְתִיב (שם פסוק כא): ״וַיֹּאמֶר אֶלְעָזָר הַכֹּהֵן אֶל אַנְשֵׁי הַצָּבָא הַבָּאִים לַמִּלְחָמָה, זֹאת חֻקַּת הַתּוֹרָה״. מְלַמֵּד שֶׁנִּשְׁתַּכְּחָה הֲלָכָה מִמֹּשֶׁה (ויקרא רבה יג א). וְעַתָּה הָבֵן: אִם כָּךְ הִגִּיעַ לְמֹשֶׁה רַבֵּינוּ עָלָיו הַשָּׁלוֹם, מָה יַגִּיעַ לַכְּסִילִים הַכּוֹעֲסִים? וּלְכָךְ אָמַר שְׁלֹמֹה (קהלת ז ט): ״אַל תְּבַהֵל בְּרוּחֲךָ לִכְעוֹס״.
Anger leads to mistakes. Who is a greater man than Moses, our teacher? Moses, upon him be peace, was angry in three places, and he made what would generally be termed "mistakes". As it is said : "And he was angry with Eleazar and with Ithamar" (Lev. 10:16), and it is written : "Why did you not eat the sin-offering in the sacred area?" (Lev. 10:17). And it is said : "Listen, you rebels" (Num. 20:10) and it is said : "And he struck the rock" (Ib. : 11). And it is said : "Moses became angry with the commanders of the army" (Num. 31:14), and it is written : "And Eleazar the priest said to the troops who had taken part in the fighting This is the ritual law' " (Num. 31:21). Which teaches that Moses forgot the law (while angry) (Leviticus Rabbah 13:1). And so, you can understand that if these things happened to Moses peace be upon him, when he was angry, what can happen to fools who are angry! And therefore Solomon said, "Be not hasty in thy spirit to be angry" (Eccl. 7:9).
וּֽלְהַבְדִּ֔יל בֵּ֥ין הַקֹּ֖דֶשׁ וּבֵ֣ין הַחֹ֑ל וּבֵ֥ין הַטָּמֵ֖א וּבֵ֥ין הַטָּהֽוֹר׃ וּלְהוֹרֹ֖ת אֶת־בְּנֵ֣י יִשְׂרָאֵ֑ל אֵ֚ת כׇּל־הַ֣חֻקִּ֔ים אֲשֶׁ֨ר דִּבֶּ֧ר יְהֹוָ֛ה אֲלֵיהֶ֖ם בְּיַד־מֹשֶֽׁה׃ {פ}
for you must distinguish between the sacred and the profane, and between the impure and the pure; and you must teach the Israelites all the laws which יהוה has imparted to them through Moses.
וזאת התורה התמימה המשלמת אותנו להיות האדם טבעי הולך בדרך האמצעיה, יאכל מה שיש לו לאכול בשויי, וישתה מה שיש לו לשתות בשווי, ויבעול מה שמותר לו לבעול בשווי, וישכון המדינות ביושר ואמונה לא שישכון במדברות ובהרים, ולא שילבש השער והצמר ולא שיענה גופו, והזהירה מזה לפי מה שבא.... שאם יאמרו אלו המתדמים באומות מאנשי תורתינו, שאיני מדבר כי אם בהם, שהם אינם עושים מה שעושים אותו מהטריח גופותם ופסוק הנאותיהם אלא על דרך הלמוד לכחות הנפש, כדי שיהיו נוטים אל הצד האחד מעט כפי מה שבארנו בזה הפרק שראוי שיהיה האדם כן, זהו טעות מהם כאשר אבאר.
The perfect Law which leads us to perfection...aims at man's following the path of moderation, in accordance with the dictates of nature, eating, drinking, enjoying legitimate sexual intercourse, all in moderation, and living among people in honesty and uprightness, but not dwelling in the wilderness or in the mountains, or clothing oneself in garments of hair and wool, or afflicting the body.....By the words of our prophets and of the sages of our Law, we see that they were bent upon moderation and the care of their souls and bodies, in accordance with what the Law prescribes mercy every man to his brother. ~The Eight Chapters of Maimonides, by Joseph I. Gorfinkle.
(1) יהוה spoke to Moses and Aaron, saying to them: (2) Speak to the Israelite people thus: These are the creatures that you may eat from among all the land animals: (3) any animal that has true hoofs, with clefts through the hoofs, and that chews the cud—such you may eat. (4) The following, however, of those that either chew the cud or have true hoofs, you shall not eat: the camel—although it chews the cud, it has no true hoofs: it is impure for you; (5) the daman—although it chews the cud, it has no true hoofs: it is impure for you; (6) the hare—although it chews the cud, it has no true hoofs: it is impure for you; (7) and the swine—although it has true hoofs, with the hoofs cleft through, it does not chew the cud: it is impure for you. (8) You shall not eat of their flesh or touch their carcasses; they are impure for you.
(9) These you may eat of all that live in water: anything in water, whether in the seas or in the streams, that has fins and scales—these you may eat. (10) But anything in the seas or in the streams that has no fins and scales, among all the swarming things of the water and among all the other living creatures that are in the water—they are an abomination for you (11) and an abomination for you they shall remain: you shall not eat of their flesh and you shall abominate their carcasses. (12) Everything in water that has no fins and scales shall be an abomination for you.
(13) The following you shall abominate among the birds—they shall not be eaten, they are an abomination: the eagle, the vulture, and the black vulture;
(מד) כִּ֣י אֲנִ֣י יְהֹוָה֮ אֱלֹֽהֵיכֶם֒ וְהִתְקַדִּשְׁתֶּם֙ וִהְיִיתֶ֣ם קְדֹשִׁ֔ים כִּ֥י קָד֖וֹשׁ אָ֑נִי וְלֹ֤א תְטַמְּאוּ֙ אֶת־נַפְשֹׁ֣תֵיכֶ֔ם בְּכׇל־הַשֶּׁ֖רֶץ הָרֹמֵ֥שׂ עַל־הָאָֽרֶץ׃ (מה) כִּ֣י ׀ אֲנִ֣י יְהֹוָ֗ה הַֽמַּעֲלֶ֤ה אֶתְכֶם֙ מֵאֶ֣רֶץ מִצְרַ֔יִם לִהְיֹ֥ת לָכֶ֖ם לֵאלֹהִ֑ים וִהְיִיתֶ֣ם קְדֹשִׁ֔ים כִּ֥י קָד֖וֹשׁ אָֽנִי׃ (מו) זֹ֣את תּוֹרַ֤ת הַבְּהֵמָה֙ וְהָע֔וֹף וְכֹל֙ נֶ֣פֶשׁ הַֽחַיָּ֔ה הָרֹמֶ֖שֶׂת בַּמָּ֑יִם וּלְכׇל־נֶ֖פֶשׁ הַשֹּׁרֶ֥צֶת עַל־הָאָֽרֶץ׃ (מז) לְהַבְדִּ֕יל בֵּ֥ין הַטָּמֵ֖א וּבֵ֣ין הַטָּהֹ֑ר וּבֵ֤ין הַֽחַיָּה֙ הַֽנֶּאֱכֶ֔לֶת וּבֵין֙ הַֽחַיָּ֔ה אֲשֶׁ֖ר לֹ֥א תֵאָכֵֽל׃ {פ}
(44) For I יהוה am your God: you shall sanctify yourselves and be holy, for I am holy. You shall not make yourselves impure through any swarming thing that moves upon the earth. (45) For I יהוה am the One who brought you up from the land of Egypt to be your God: you shall be holy, for I am holy. (46) These are the instructions concerning animals, birds, all living creatures that move in water, and all creatures that swarm on earth, (47) for distinguishing between the impure and the pure, between the living things that may be eaten and the living things that may not be eaten.
